Derivative

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The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

Differentiation

Differentiation is the action of computing a derivative. The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x. If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point.

Slope of a linear function: ${\displaystyle m={\frac {\Delta y}{\Delta x}}}$

The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y is a line. In this case, y = f(x) = mx + b, for real numbers m and b, and the slope m is given by

${\displaystyle m={\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {\Delta y}{\Delta x}},}$

where the symbol Δ (Delta) is an abbreviation for "change in". This formula is true because

${\displaystyle y+\Delta y=f\left(x+\Delta x\right)=m\left(x+\Delta x\right)+b=mx+m\,\Delta x+b=y+m\,\Delta x.}$

Thus, since

${\displaystyle y+\Delta y=y+m\,\Delta x,}$

it follows that

${\displaystyle \Delta y=m\,\Delta x.}$

This gives an exact value for the slope of a line. If the function f is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

Rate of change as a limit value

Figure 1. The tangent line at (x, f(x))

Figure 2. The secant to curve y= f(x) determined by points (x, f(x)) and (x + h, f(x + h))

Figure 3. The tangent line as limit of secants

Figure 4. Animated illustration: the tangent line (derivative) as the limit of secants

The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.

Notation

Two distinct notations are commonly used for the derivative, one deriving from Leibniz and the other from Joseph Louis Lagrange.

In Leibniz's notation, an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written

${\displaystyle {\frac {dy}{dx}}}$

suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of y with respect to x", "dy by dx", or "dy over dx". The oral form "dy dx" is often used conversationally, although it may lead to confusion.)

In Lagrange's notation, the derivative with respect to x of a function f(x) is denoted f'(x) (read as "f prime of x") or f_x′(x) (read as "f prime x of x"), in case of ambiguity of the variable implied by the derivation. Lagrange's notation is sometimes incorrectly attributed to Newton.

Rigorous definition

A secant approaches a tangent when ${\displaystyle \Delta x\to 0}$.

The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers. This is the approach described below.

Let f be a real valued function defined in an open neighborhood of a real number a. In classical geometry, the tangent line to the graph of the function f at a was the unique line through the point (a, f(a)) that did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)). The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant line is the difference between the y values of these points divided by the difference between the x values, that is,

${\displaystyle m={\frac {\Delta f(a)}{\Delta a}}={\frac {f(a+h)-f(a)}{(a+h)-(a)}}={\frac {f(a+h)-f(a)}{h}}.}$

This expression is Newton's difference quotient. Passing from an approximation to an exact answer is done using a limit. Geometrically, the limit of the secant lines is the tangent line. Therefore, the limit of the difference quotient as h approaches zero, if it exists, should represent the slope of the tangent line to (a, f(a)). This limit is defined to be the derivative of the function f at a:

${\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.}$

When the limit exists, f is said to be differentiable at a. Here f′(a) is one of several common notations for the derivative (see below).

Equivalently, the derivative satisfies the property that

${\displaystyle \lim _{h\to 0}{\frac {f(a+h)-(f(a)+f'(a)\cdot h)}{h}}=0,}$

which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation

${\displaystyle f(a+h)\approx f(a)+f'(a)h}$

to f near a (i.e., for small h). This interpretation is the easiest to generalize to other settings (see below).

Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly using this method. Instead, define Q(h) to be the difference quotient as a function of h:

${\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.}$

Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from h = 0. If the limit lim_h→0Q(h) exists, meaning that there is a way of choosing a value for Q(0) that makes Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q(0).

In practice, the existence of a continuous extension of the difference quotient Q(h) to h = 0 is shown by modifying the numerator to cancel h in the denominator. Such manipulations can make the limit value of Q for small h clear even though Q is still not defined at h = 0. This process can be long and tedious for complicated functions, and many shortcuts are commonly used to simplify the process.

Definition over the hyperreals

Relative to a hyperreal extension R ⊂ ^∗R of the real numbers, the derivative of a real function y = f(x) at a real point x can be defined as the shadow of the quotient ∆y/∆x for infinitesimal ∆x, where ∆y = f(x + ∆x) − f(x). Here the natural extension of f to the hyperreals is still denoted f. Here the derivative is said to exist if the shadow is independent of the infinitesimal chosen.

Example

The squaring function

The squaring function given by f(x) = x² is differentiable at x = 3, and its derivative there is 6. This result is established by calculating the limit as h approaches zero of the difference quotient of f(3):

${\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{\frac {f(3+h)-f(3)}{h}}=\lim _{h\to 0}{\frac {(3+h)^{2}-3^{2}}{h}}\\[10pt]&=\lim _{h\to 0}{\frac {9+6h+h^{2}-9}{h}}=\lim _{h\to 0}{\frac {6h+h^{2}}{h}}=\lim _{h\to 0}{(6+h)}.\end{aligned}}}$

The last expression shows that the difference quotient equals 6 + h when h ≠ 0 and is undefined when h = 0, because of the definition of the difference quotient. However, the definition of the limit says the difference quotient does not need to be defined when h = 0. The limit is the result of letting h go to zero, meaning it is the value that 6 + h tends to as h becomes very small:

${\displaystyle \lim _{h\to 0}{(6+h)}=6+0=6.}$

Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at x = 3 is f′(3) = 6.

More generally, a similar computation shows that the derivative of the squaring function at x = a is f′(a) = 2a:

${\displaystyle {\begin{aligned}f'(a)&=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}=\lim _{h\to 0}{\frac {(a+h)^{2}-a^{2}}{h}}\\[0.3em]&=\lim _{h\to 0}{\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=\lim _{h\to 0}{\frac {2ah+h^{2}}{h}}\\[0.3em]&=\lim _{h\to 0}{(2a+h)}=2a\end{aligned}}}$

Continuity and differentiability

This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity).

If f is differentiable at a, then f must also be continuous at a. As an example, choose a point a and let f be the step function that returns the value 1 for all x less than a, and returns a different value 10 for all x greater than or equal to a. f cannot have a derivative at a. If h is negative, then a + h is on the low part of the step, so the secant line from a to a + h is very steep, and as h tends to zero the slope tends to infinity. If h is positive, then a + h is on the high part of the step, so the secant line from a to a + h has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.

The absolute value function is continuous, but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do from the right.

However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by f(x) = |x| is continuous at x = 0, but it is not differentiable there. If h is positive, then the slope of the secant line from 0 to h is one, whereas if h is negative, then the slope of the secant line from 0 to h is negative one. This can be seen graphically as a "kink" or a "cusp" in the graph at x = 0. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by f(x) = x^1/3 is not differentiable at x = 0.

In summary: for a function f to have a derivative it is necessary for the function f to be continuous, but continuity alone is not sufficient.

Most functions that occur in practice have derivatives at all points or at almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, for example if the function is a monotone function or a Lipschitz function, this is true. However, in 1872 Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. Informally, this means that hardly do any random continuous functions have a derivative at even one point.

The derivative as a function

The derivative at different points of a differentiable function. In this case, the derivative is equal to:${\displaystyle \sin(x^{2})+2x^{2}\cos(x^{2})}$

Let f be a function that has a derivative at every point in its domain. We can then define a function that maps every point ${\displaystyle x}$ to the value of the derivative of ${\displaystyle f}$ at ${\displaystyle x}$. This function is written f′ and is called the derivative function or the derivative of f.

Sometimes f has a derivative at most, but not all, points of its domain. The function whose value at a equals f′(a) whenever f′(a) is defined and elsewhere is undefined is also called the derivative of f. It is still a function, but its domain is strictly smaller than the domain of f.

Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by D, then D(f) is the function f′. Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f′(a).

For comparison, consider the doubling function given by f(x) = 2x; f is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs:

${\displaystyle {\begin{aligned}1&{}\mapsto 2,\\2&{}\mapsto 4,\\3&{}\mapsto 6.\end{aligned}}}$

The operator D, however, is not defined on individual numbers. It is only defined on functions:

${\displaystyle {\begin{aligned}D(x\mapsto 1)&=(x\mapsto 0),\\D(x\mapsto x)&=(x\mapsto 1),\\D(x\mapsto x^{2})&=(x\mapsto 2\cdot x).\end{aligned}}}$

Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to the squaring function, x ↦ x², D outputs the doubling function x ↦ 2x, which we named f(x). This output function can then be evaluated to get f(1) = 2, f(2) = 4, and so on.

Higher derivatives

Let f be a differentiable function, and let f ′ be its derivative. The derivative of f ′ (if it has one) is written f ′′ and is called the second derivative of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ and is called the third derivative of f. Continuing this process, one can define, if it exists, the nth derivative as the derivative of the (n-1)th derivative. These repeated derivatives are called higher-order derivatives. The nth derivative is also called the derivative of order n.

If x(t) represents the position of an object at time t, then the higher-order derivatives of x have physical interpretations. The second derivative of x is the derivative of x′, the velocity, and by definition this is the object's acceleration. The third derivative of x is defined to be the jerk, and the fourth derivative is defined to be the jounce.

A function f need not have a derivative (for example, if it is not continuous). Similarly, even if f does have a derivative, it may not have a second derivative. For example, let

${\displaystyle f(x)={\begin{cases}+x^{2},&{\text{if }}x\geq 0\\-x^{2},&{\text{if }}x\leq 0.\end{cases}}}$

Calculation shows that f is a differentiable function whose derivative at ${\displaystyle x}$ is given by

${\displaystyle f'(x)={\begin{cases}+2x,&{\text{if }}x\geq 0\\-2x,&{\text{if }}x\leq 0.\end{cases}}}$

f'(x) is twice the absolute value function at ${\displaystyle x}$, and it does not have a derivative at zero. Similar examples show that a function can have a kth derivative for each non-negative integer k but not a (k + 1)th derivative. A function that has k successive derivatives is called k times differentiable. If in addition the kth derivative is continuous, then the function is said to be of differentiability class C^k. A function that has infinitely many derivatives is called infinitely differentiable or smooth.

On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.

The derivatives of a function f at a point x provide polynomial approximations to that function near x. For example, if f is twice differentiable, then

${\displaystyle f(x+h)\approx f(x)+f'(x)h+{\tfrac {1}{2}}f''(x)h^{2}}$

in the sense that

${\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)-f'(x)h-{\frac {1}{2}}f''(x)h^{2}}{h^{2}}}=0.}$

If f is infinitely differentiable, then this is the beginning of the Taylor series for f evaluated at x + h around x.

Inflection point

A point where the second derivative of a function changes sign is called an inflection point. At an inflection point, the second derivative may be zero, as in the case of the inflection point x = 0 of the function given by ${\displaystyle f(x)=x^{3}}$, or it may fail to exist, as in the case of the inflection point x = 0 of the function given by ${\displaystyle f(x)=x^{\frac {1}{3}}}$. At an inflection point, a function switches from being a convex function to being a concave function or vice versa.

Notation (details)

Leibniz's notation

The symbols ${\displaystyle dx}$, ${\displaystyle dy}$, and ${\displaystyle {\frac {dy}{dx}}}$ were introduced by Gottfried Wilhelm Leibniz in 1675. It is still commonly used when the equation y = f(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by

${\displaystyle {\frac {dy}{dx}},\quad {\frac {df}{dx}},{\text{ or }}{\frac {d}{dx}}f,}$

and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation

${\displaystyle {\frac {d^{n}y}{dx^{n}}},\quad {\frac {d^{n}f}{dx^{n}}},{\text{ or }}{\frac {d^{n}}{dx^{n}}}f}$

for the nth derivative of ${\displaystyle y=f(x)}$. These are abbreviations for multiple applications of the derivative operator. For example,

${\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right).}$

With Leibniz's notation, we can write the derivative of ${\displaystyle y}$ at the point ${\displaystyle x=a}$ in two different ways:

${\displaystyle \left.{\frac {dy}{dx}}\right|_{x=a}={\frac {dy}{dx}}(a).}$

Leibniz's notation allows one to specify the variable for differentiation (in the denominator), which is relevant in partial differentiation. It also makes the chain rule easier to remember:

${\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}$

Lagrange's notation

Sometimes referred to as prime notation, one of the most common modern notation for differentiation is due to Joseph-Louis Lagrange and uses the prime mark, so that the derivative of a function ${\displaystyle f}$ is denoted ${\displaystyle f'}$. Similarly, the second and third derivatives are denoted

${\displaystyle (f')'=f''}$   and   ${\displaystyle (f'')'=f'''.}$

To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses:

${\displaystyle f^{\mathrm {iv} }}$   or   ${\displaystyle f^{(4)}.}$

The latter notation generalizes to yield the notation ${\displaystyle f^{(n)}}$ for the nth derivative of ${\displaystyle f}$ – this notation is most useful when we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can become cumbersome.

Newton's notation

Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If ${\displaystyle y=f(t)}$, then

${\displaystyle {\dot {y}}}$   and   ${\displaystyle {\ddot {y}}}$

denote, respectively, the first and second derivatives of ${\displaystyle y}$. This notation is used exclusively for derivatives with respect to time or arc length. It is very common in physics, differential equations, and differential geometry. While the notation becomes unmanageable for high-order derivatives, in practice only few derivatives are needed.

Euler's notation

Euler's notation uses a differential operator ${\displaystyle D}$, which is applied to a function ${\displaystyle f}$ to give the first derivative ${\displaystyle Df}$. The nth derivative is denoted ${\displaystyle D^{n}f}$.

If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written

${\displaystyle D_{x}y}$   or   ${\displaystyle D_{x}f(x)}$,

although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression.

Euler's notation is useful for stating and solving linear differential equations.

Rules of computation

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.

Rules for basic functions

Most derivative computations eventually require taking the derivative of some common functions. The following incomplete list gives some of the most frequently used functions of a single real variable and their derivatives.

• Derivatives of powers: if

${\displaystyle f(x)=x^{r},}$

where r is any real number, then

${\displaystyle f'(x)=rx^{r-1},}$

wherever this is defined. For example, if ${\displaystyle f(x)=x^{1/4}}$, then

${\displaystyle f'(x)=(1/4)x^{-3/4},}$

and the derivative function is defined only for positive x, not for x = 0. When r = 0, this rule implies that f′(x) is zero for x ≠ 0, which is almost the constant rule (stated below).

• Exponential and logarithmic functions:

${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}$

${\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a).}$

${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad x>0.}$

${\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}.}$

• Trigonometric functions:

${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x).}$

${\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x).}$

${\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x).}$

• Inverse trigonometric functions:

${\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1$

${\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1$

${\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}$

Rules for combined functions

In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules. Some of the most basic rules are the following.

• Constant rule: if f(x) is constant, then

${\displaystyle f'(x)=0.}$

• Sum rule:

${\displaystyle (\alpha f+\beta g)'=\alpha f'+\beta g'}$ for all functions f and g and all real numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.

• Product rule:

${\displaystyle (fg)'=f'g+fg'}$ for all functions f and g. As a special case, this rule includes the fact ${\displaystyle (\alpha f)'=\alpha f'}$ whenever ${\displaystyle \alpha }$ is a constant, because ${\displaystyle \alpha 'f=0\cdot f=0}$ by the constant rule.

• Quotient rule:

${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}$ for all functions f and g at all inputs where g ≠ 0.

• Chain rule: If ${\displaystyle f(x)=h(g(x))}$, then

${\displaystyle f'(x)=h'(g(x))\cdot g'(x).}$

Computation example

The derivative of the function given by

${\displaystyle f(x)=x^{4}+\sin(x^{2})-\ln(x)e^{x}+7}$

is

${\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos(x^{2})-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln(x){\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos(x^{2})-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}}$

Here the second term was computed using the chain rule and third using the product rule. The known derivatives of the elementary functions x², x⁴, sin(x), ln(x) and exp(x) = e^x, as well as the constant 7, were also used.

In higher dimensions

Vector-valued functions

A vector-valued function y of a real variable sends real numbers to vectors in some vector space Rⁿ. A vector-valued function can be split up into its coordinate functions y₁(t), y₂(t), …, y_n(t), meaning that y(t) = (y₁(t), ..., y_n(t)). This includes, for example, parametric curves in R² or R³. The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,

${\displaystyle \mathbf {y} '(t)=(y'_{1}(t),\ldots ,y'_{n}(t)).}$

Equivalently,

${\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},}$

if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of y exists for every value of t, then y′ is another vector-valued function.

If e₁, …, e_n is the standard basis for Rⁿ, then y(t) can also be written as y₁(t)e₁ + … + y_n(t)e_n. If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be

${\displaystyle y'_{1}(t)\mathbf {e} _{1}+\cdots +y'_{n}(t)\mathbf {e} _{n}}$

because each of the basis vectors is a constant.

This generalization is useful, for example, if y(t) is the position vector of a particle at time t; then the derivative y′(t) is the velocity vector of the particle at time t.

Partial derivatives

Suppose that f is a function that depends on more than one variable—for instance,

${\displaystyle f(x,y)=x^{2}+xy+y^{2}.}$

f can be reinterpreted as a family of functions of one variable indexed by the other variables:

${\displaystyle f(x,y)=f_{x}(y)=x^{2}+xy+y^{2}.}$

In other words, every value of x chooses a function, denoted f_x, which is a function of one real number. That is,

${\displaystyle x\mapsto f_{x},}$

${\displaystyle f_{x}(y)=x^{2}+xy+y^{2}.}$

Once a value of x is chosen, say a, then f(x, y) determines a function f_a that sends y to a² + ay + y²:

${\displaystyle f_{a}(y)=a^{2}+ay+y^{2}.}$

In this expression, a is a constant, not a variable, so f_a is a function of only one real variable. Consequently, the definition of the derivative for a function of one variable applies:

${\displaystyle f_{a}'(y)=a+2y.}$

The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function that describes the variation of f in the y direction:

${\displaystyle {\frac {\partial f}{\partial y}}(x,y)=x+2y.}$

This is the partial derivative of f with respect to y. Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".

In general, the partial derivative of a function f(x₁, …, x_n) in the direction x_i at the point (a₁, …, a_n) is defined to be:

${\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\ldots ,a_{n})}{h}}.}$

In the above difference quotient, all the variables except x_i are held fixed. That choice of fixed values determines a function of one variable

${\displaystyle f_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}(x_{i})=f(a_{1},\ldots ,a_{i-1},x_{i},a_{i+1},\ldots ,a_{n}),}$

and, by definition,

${\displaystyle {\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{i})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n}).}$

In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

An important example of a function of several variables is the case of a scalar-valued function f(x₁, ..., x_n) on a domain in Euclidean space Rⁿ (e.g., on R² or R³). In this case f has a partial derivative ∂f/∂x_j with respect to each variable x_j. At the point (a₁, …, a_n), these partial derivatives define the vector

${\displaystyle \nabla f(a_{1},\ldots ,a_{n})=\left({\frac {\partial f}{\partial x_{1}}}(a_{1},\ldots ,a_{n}),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a_{1},\ldots ,a_{n})\right).}$

This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f that takes the point (a₁, …, a_n) to the vector ∇f(a₁, …, a_n). Consequently, the gradient determines a vector field.

Directional derivatives

If f is a real-valued function on Rⁿ, then the partial derivatives of f measure its variation in the direction of the coordinate axes. For example, if f is a function of x and y, then its partial derivatives measure the variation in f in the x direction and the y direction. They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x. These are measured using directional derivatives. Choose a vector

${\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n}).}$

The directional derivative of f in the direction of v at the point x is the limit

${\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}$

In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector. Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. To see how this works, suppose that v = λu. Substitute h = k/λ into the difference quotient. The difference quotient becomes:

${\displaystyle {\frac {f(\mathbf {x} +(k/\lambda )(\lambda \mathbf {u} ))-f(\mathbf {x} )}{k/\lambda }}=\lambda \cdot {\frac {f(\mathbf {x} +k\mathbf {u} )-f(\mathbf {x} )}{k}}.}$

This is λ times the difference quotient for the directional derivative of f with respect to u. Furthermore, taking the limit as h tends to zero is the same as taking the limit as k tends to zero because h and k are multiples of each other. Therefore, D_v(f) = λD_u(f). Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.

If all the partial derivatives of f exist and are continuous at x, then they determine the directional derivative of f in the direction v by the formula:

${\displaystyle D_{\mathbf {v} }{f}({\boldsymbol {x}})=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.}$

This is a consequence of the definition of the total derivative. It follows that the directional derivative is linear in v, meaning that D_{v + w}(f) = D_v(f) + D_w(f).

The same definition also works when f is a function with values in R^m. The above definition is applied to each component of the vectors. In this case, the directional derivative is a vector in R^m.

Total derivative, total differential and Jacobian matrix

When f is a function from an open subset of Rⁿ to R^m, then the directional derivative of f in a chosen direction is the best linear approximation to f at that point and in that direction. But when n > 1, no single directional derivative can give a complete picture of the behavior of f. The total derivative gives a complete picture by considering all directions at once. That is, for any vector v starting at a, the linear approximation formula holds:

${\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .}$

Just like the single-variable derivative, f ′(a) is chosen so that the error in this approximation is as small as possible.

If n and m are both one, then the derivative f ′(a) is a number and the expression f ′(a)v is the product of two numbers. But in higher dimensions, it is impossible for f ′(a) to be a number. If it were a number, then f ′(a)v would be a vector in Rⁿ while the other terms would be vectors in R^m, and therefore the formula would not make sense. For the linear approximation formula to make sense, f ′(a) must be a function that sends vectors in Rⁿ to vectors in R^m, and f ′(a)v must denote this function evaluated at v.

To determine what kind of function it is, notice that the linear approximation formula can be rewritten as

${\displaystyle f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} )\approx f'(\mathbf {a} )\mathbf {v} .}$

Notice that if we choose another vector w, then this approximate equation determines another approximate equation by substituting w for v. It determines a third approximate equation by substituting both w for v and a + v for a. By subtracting these two new equations, we get

${\displaystyle f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\approx f'(\mathbf {a} +\mathbf {v} )\mathbf {w} -f'(\mathbf {a} )\mathbf {w} .}$

If we assume that v is small and that the derivative varies continuously in a, then f ′(a + v) is approximately equal to f ′(a), and therefore the right-hand side is approximately zero. The left-hand side can be rewritten in a different way using the linear approximation formula with v + w substituted for v. The linear approximation formula implies:

${\displaystyle {\begin{aligned}0&\approx f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\\&=(f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {w} )-f(\mathbf {a} ))\\&\approx f'(\mathbf {a} )(\mathbf {v} +\mathbf {w} )-f'(\mathbf {a} )\mathbf {v} -f'(\mathbf {a} )\mathbf {w} .\end{aligned}}}$

This suggests that f ′(a) is a linear transformation from the vector space Rⁿ to the vector space R^m. In fact, it is possible to make this a precise derivation by measuring the error in the approximations. Assume that the error in these linear approximation formula is bounded by a constant times ||v||, where the constant is independent of v but depends continuously on a. Then, after adding an appropriate error term, all of the above approximate equalities can be rephrased as inequalities. In particular, f ′(a) is a linear transformation up to a small error term. In the limit as v and w tend to zero, it must therefore be a linear transformation. Since we define the total derivative by taking a limit as v goes to zero, f ′(a) must be a linear transformation.

In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors. In particular, the numerator and denominator of the difference quotient are not even in the same vector space: The numerator lies in the codomain R^m while the denominator lies in the domain Rⁿ. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator and denominator. To make precise the idea that f ′(a) is the best linear approximation, it is necessary to adapt a different formula for the one-variable derivative in which these problems disappear. If f : R → R, then the usual definition of the derivative may be manipulated to show that the derivative of f at a is the unique number f ′(a) such that

${\displaystyle \lim _{h\to 0}{\frac {f(a+h)-(f(a)+f'(a)h)}{h}}=0.}$

This is equivalent to

${\displaystyle \lim _{h\to 0}{\frac {|f(a+h)-(f(a)+f'(a)h)|}{|h|}}=0}$

because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero. This last formula can be adapted to the many-variable situation by replacing the absolute values with norms.

The definition of the total derivative of f at a, therefore, is that it is the unique linear transformation f ′(a) : Rⁿ → R^m such that

${\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-(f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {h} )\rVert }{\lVert \mathbf {h} \rVert }}=0.}$

Here h is a vector in Rⁿ, so the norm in the denominator is the standard length on Rⁿ. However, f′(a)h is a vector in R^m, and the norm in the numerator is the standard length on R^m. If v is a vector starting at a, then f ′(a)v is called the pushforward of v by f and is sometimes written f_∗v.

If the total derivative exists at a, then all the partial derivatives and directional derivatives of f exist at a, and for all v, f ′(a)v is the directional derivative of f in the direction v. If we write f using coordinate functions, so that f = (f₁, f₂, ..., f_m), then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of f at a:

${\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}$

The existence of the total derivative f′(a) is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on a.

The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if f is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative f′(x). This 1×1 matrix satisfies the property that f(a + h) − (f(a) + f ′(a)h) is approximately zero, in other words that

${\displaystyle f(a+h)\approx f(a)+f'(a)h.}$

Up to changing variables, this is the statement that the function ${\displaystyle x\mapsto f(a)+f'(a)(x-a)}$ is the best linear approximation to f at a.

The total derivative of a function does not give another function in the same way as the one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target.

The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a jet, cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors. It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the jet bundle. The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the kth order jet of a function and its partial derivatives of order less than or equal to k.

By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to R^p. The kth order total derivative may be interpreted as a map

${\displaystyle D^{k}f:\mathbb {R} ^{n}\to L^{k}(\mathbb {R} ^{n}\times \cdots \times \mathbb {R} ^{n},\mathbb {R} ^{m})}$

which takes a point x in Rⁿ and assigns to it an element of the space of k-linear maps from Rⁿ to R^m – the "best" (in a certain precise sense) k-linear approximation to f at that point. By precomposing it with the diagonal map Δ, x → (x, x), a generalized Taylor series may be begun as

${\displaystyle {\begin{aligned}f(\mathbf {x} )&\approx f(\mathbf {a} )+(Df)(\mathbf {x-a} )+(D^{2}f)(\Delta (\mathbf {x-a} ))+\cdots \\&=f(\mathbf {a} )+(Df)(\mathbf {x-a} )+(D^{2}f)(\mathbf {x-a} ,\mathbf {x-a} )+\cdots \\&=f(\mathbf {a} )+\sum _{i}(Df)_{i}(x_{i}-a_{i})+\sum _{j,k}(D^{2}f)_{jk}(x_{j}-a_{j})(x_{k}-a_{k})+\cdots \end{aligned}}}$

where f(a) is identified with a constant function, x_i − a_i are the components of the vector x − a, and (Df)_i and (D²f)_jk are the components of Df and D²f as linear transformations.

Generalizations

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point.

• An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. If C is identified with R² by writing a complex number z as x + iy, then a differentiable function from C to C is certainly differentiable as a function from R² to R² (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the Cauchy–Riemann equations – see holomorphic functions.
• Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space that can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R³. The derivative (or differential) of a (differentiable) map f: M → N between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses – see pushforward (differential) and pullback (differential geometry).
• Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gâteaux derivative, and of the differential, called the Fréchet derivative.
• One deficiency of the classical derivative is that not very many functions are differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all continuous functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average".
• The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra.
• The discrete equivalent of differentiation is finite differences. The study of differential calculus is unified with the calculus of finite differences in time scale calculus.

History of calculus

From Wikipedia, the free encyclopedia

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century. However, each inventor claimed the other stole his work in a bitter dispute that continued until the end of their lives.

Precursors of calculus

Ancient

Archimedes used the method of exhaustion to compute the area inside a circle

The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. Babylonians may have discovered the trapezoidal rule while doing astronomical observations of Jupiter.

From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create.

Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in his The Quadrature of the Parabola, The Method, and On the Sphere and Cylinder. It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th century that the method was formalized by Cavalieri as the method of Indivisibles and eventually incorporated by Newton into a general framework of integral calculus. Archimedes was the first to find the tangent to a curve other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve. The pioneers of the calculus such as Isaac Barrow and Johann Bernoulli were diligent students of Archimedes; see for instance C. S. Roero (1983).

Medieval

The method of exhaustion was reinvented in China by Liu Hui in the 4th century AD in order to find the area of a circle.^[7] In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. In the 14th century, Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics stated components of calculus such as the Taylor series and infinite series approximations. However, they were not able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the powerful problem-solving tool we have today.

The mathematical study of continuity was revived in the 14th century by the Oxford Calculators and French collaborators such as Nicole Oresme. They proved the "Merton mean speed theorem": that a uniformly accelerated body travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body.

Pioneers of modern calculus

In the 17th century, European mathematicians Isaac Barrow, René Descartes, Pierre de Fermat, Blaise Pascal, John Wallis and others discussed the idea of a derivative. In particular, in Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed an adequality method for determining maxima, minima, and tangents to various curves that was closely related to differentiation. Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."

On the integral side, Cavalieri developed his method of indivisibles in the 1630s and 1640s, providing a more modern form of the ancient Greek method of exhaustion, and computing Cavalieri's quadrature formula, the area under the curves xⁿ of higher degree, which had previously only been computed for the parabola, by Archimedes. Torricelli extended this work to other curves such as the cycloid, and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. James Gregory, influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus in the mid-17th century. The first full proof of the fundamental theorem of calculus was given by Isaac Barrow.

Newton and Leibniz, building on this work, independently developed the surrounding theory of infinitesimal calculus in the late 17th century. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton provided some of the most important applications to physics, especially of integral calculus.

The first proof of Rolle's theorem was given by Michel Rolle in 1691 using methods developed by the Dutch mathematician Johann van Waveren Hudde. The mean value theorem in its modern form was stated by Bernard Bolzano and Augustin-Louis Cauchy (1789–1857) also after the founding of modern calculus. Important contributions were also made by Barrow, Huygens, and many others.

Newton and Leibniz

Isaac Newton

 

Gottfried Leibniz

Before Newton and Leibniz, the word “calculus” referred to any body of mathematics, but in the following years, "calculus" became a popular term for a field of mathematics based upon their insights. The purpose of this section is to examine Newton and Leibniz’s investigations into the developing field of infinitesimal calculus. Specific importance will be put on the justification and descriptive terms which they used in an attempt to understand calculus as they themselves conceived it.

By the middle of the 17th century, European mathematics had changed its primary repository of knowledge. In comparison to the last century which maintained Hellenistic mathematics as the starting point for research, Newton, Leibniz and their contemporaries increasingly looked towards the works of more modern thinkers. Europe had become home to a burgeoning mathematical community and with the advent of enhanced institutional and organizational bases a new level of organization and academic integration was being achieved. Importantly, however, the community lacked formalism; instead it consisted of a disordered mass of various methods, techniques, notations, theories, and paradoxes.

Newton came to calculus as part of his investigations in physics and geometry. He viewed calculus as the scientific description of the generation of motion and magnitudes. In comparison, Leibniz focused on the tangent problem and came to believe that calculus was a metaphysical explanation of change. Importantly, the core of their insight was the formalization of the inverse properties between the integral and the differential of a function. This insight had been anticipated by their predecessors, but they were the first to conceive calculus as a system in which new rhetoric and descriptive terms were created. Their unique discoveries lay not only in their imagination, but also in their ability to synthesize the insights around them into a universal algorithmic process, thereby forming a new mathematical system.

Newton

Newton completed no definitive publication formalizing his fluxional calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as the Principia and Opticks. Newton would begin his mathematical training as the chosen heir of Isaac Barrow in Cambridge. His aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing the binomial theorem, which he had extended to include fractional and negative exponents. Newton succeeded in expanding the applicability of the binomial theorem by applying the algebra of finite quantities in an analysis of infinite series. He showed a willingness to view infinite series not only as approximate devices, but also as alternative forms of expressing a term.

Many of Newton's critical insights occurred during the plague years of 1665–1666 which he later described as, "the prime of my age for invention and minded mathematics and [natural] philosophy more than at any time since." It was during his plague-induced isolation that the first written conception of fluxionary calculus was recorded in the unpublished De Analysi per Aequationes Numero Terminorum Infinitas. In this paper, Newton determined the area under a curve by first calculating a momentary rate of change and then extrapolating the total area. He began by reasoning about an indefinitely small triangle whose area is a function of x and y. He then reasoned that the infinitesimal increase in the abscissa will create a new formula where x = x + o (importantly, o is the letter, not the digit 0). He then recalculated the area with the aid of the binomial theorem, removed all quantities containing the letter o and re-formed an algebraic expression for the area. Significantly, Newton would then “blot out” the quantities containing o because terms "multiplied by it will be nothing in respect to the rest".

At this point Newton had begun to realize the central property of inversion. He had created an expression for the area under a curve by considering a momentary increase at a point. In effect, the fundamental theorem of calculus was built into his calculations. While his new formulation offered incredible potential, Newton was well aware of its logical limitations at the time. He admits that "errors are not to be disregarded in mathematics, no matter how small" and that what he had achieved was “shortly explained rather than accurately demonstrated."

In an effort to give calculus a more rigorous explication and framework, Newton compiled in 1671 the Methodus Fluxionum et Serierum Infinitarum. In this book, Newton's strict empiricism shaped and defined his fluxional calculus. He exploited instantaneous motion and infinitesimals informally. He used math as a methodological tool to explain the physical world. The base of Newton’s revised calculus became continuity; as such he redefined his calculations in terms of continual flowing motion. For Newton, variable magnitudes are not aggregates of infinitesimal elements, but are generated by the indisputable fact of motion. As with many of his works, Newton delayed publication. Methodus Fluxionum was not published until 1736.

Newton attempted to avoid the use of the infinitesimal by forming calculations based on ratios of changes. In the Methodus Fluxionum he defined the rate of generated change as a fluxion, which he represented by a dotted letter, and the quantity generated he defined as a fluent. For example, if ${\displaystyle {x}}$ and ${\displaystyle {y}}$ are fluents, then ${\displaystyle {\dot {x}}}$ and ${\displaystyle {\dot {y}}}$ are their respective fluxions. This revised calculus of ratios continued to be developed and was maturely stated in the 1676 text De Quadratura Curvarum where Newton came to define the present day derivative as the ultimate ratio of change, which he defined as the ratio between evanescent increments (the ratio of fluxions) purely at the moment in question. Essentially, the ultimate ratio is the ratio as the increments vanish into nothingness. Importantly, Newton explained the existence of the ultimate ratio by appealing to motion:

“For by the ultimate velocity is meant that, with which the body is moved, neither before it arrives at its last place, when the motion ceases nor after but at the very instant when it arrives... the ultimate ratio of evanescent quantities is to be understood, the ratio of quantities not before they vanish, not after, but with which they vanish”

Newton developed his fluxional calculus in an attempt to evade the informal use of infinitesimals in his calculations.

Leibniz

While Newton began development of his fluxional calculus in 1665–1666 his findings did not become widely circulated until later. In the intervening years Leibniz also strove to create his calculus. In comparison to Newton who came to math at an early age, Leibniz began his rigorous math studies with a mature intellect. He was a polymath, and his intellectual interests and achievements involved metaphysics, law, economics, politics, logic, and mathematics. In order to understand Leibniz’s reasoning in calculus his background should be kept in mind. Particularly, his metaphysics which described the universe as a Monadology, and his plans of creating a precise formal logic whereby, "a general method in which all truths of the reason would be reduced to a kind of calculation."

In 1672 Leibniz met the mathematician Huygens who convinced Leibniz to dedicate significant time to the study of mathematics. By 1673 he had progressed to reading Pascal’s Traité des Sinus du Quarte Cercle and it was during his largely autodidactic research that Leibniz said "a light turned on". Like Newton, Leibniz, saw the tangent as a ratio but declared it as simply the ratio between ordinates and abscissas. He continued this reasoning to argue that the integral was in fact the sum of the ordinates for infinitesimal intervals in the abscissa; in effect, the sum of an infinite number of rectangles. From these definitions the inverse relationship or differential became clear and Leibniz quickly realized the potential to form a whole new system of mathematics. Where Newton over the course of his career used several approaches in addition to an approach using infinitesimals, Leibniz made this the cornerstone of his notation and calculus.

In the manuscripts of 25 October to 11 November 1675, Leibniz recorded his discoveries and experiments with various forms of notation. He was acutely aware of the notational terms used and his earlier plans to form a precise logical symbolism became evident. Eventually, Leibniz denoted the infinitesimal increments of abscissas and ordinates dx and dy, and the summation of infinitely many infinitesimally thin rectangles as a long s (∫ ), which became the present integral symbol ${\displaystyle \scriptstyle \int }$.

While Leibniz's notation is used by modern mathematics, his logical base was different from our current one.^{[citation needed]} Leibniz embraced infinitesimals and wrote extensively so as, “not to make of the infinitely small a mystery, as had Pascal.” According to Gilles Deleuze, Leibniz's zeroes "are nothings, but they are not absolute nothings, they are nothings respectively" (quoting Leibniz' text "Justification of the calculus of infinitesimals by the calculus of ordinary algebra"). Alternatively, he defines them as, “less than any given quantity.”^{[This quote needs a citation]} For Leibniz, the world was an aggregate of infinitesimal points and the lack of scientific proof for their existence did not trouble him. Infinitesimals to Leibniz were ideal quantities of a different type from appreciable numbers. The truth of continuity was proven by existence itself. For Leibniz the principle of continuity and thus the validity of his calculus was assured. Three hundred years after Leibniz's work, Abraham Robinson showed that using infinitesimal quantities in calculus could be given a solid foundation.

Legacy

The rise of calculus stands out as a unique moment in mathematics. Calculus is the mathematics of motion and change, and as such, its invention required the creation of a new mathematical system. Importantly, Newton and Leibniz did not create the same calculus and they did not conceive of modern calculus. While they were both involved in the process of creating a mathematical system to deal with variable quantities their elementary base was different. For Newton, change was a variable quantity over time and for Leibniz it was the difference ranging over a sequence of infinitely close values. Notably, the descriptive terms each system created to describe change was different.

Historically, there was much debate over whether it was Newton or Leibniz who first "invented" calculus. This argument, the Leibniz and Newton calculus controversy, involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of tangents by the time Leibniz became interested in the question. Much of the controversy centers on the question whether Leibniz had seen certain early manuscripts of Newton before publishing his own memoirs on the subject. Newton began his work on calculus no later than 1666, and Leibniz did not begin his work until 1673. Leibniz visited England in 1673 and again in 1676, and was shown some of Newton's unpublished writings. He also corresponded with several English scientists (as well as with Newton himself), and may have gained access to Newton's manuscripts through them. It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists at the turn of the century, but after 1711 both of them became personally involved, accusing each other of plagiarism.

The priority dispute had an effect of separating English-speaking mathematicians from those in the continental Europe for many years. Only in the 1820s, due to the efforts of the Analytical Society, did Leibnizian analytical calculus become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of fluents and fluxions".

The work of both Newton and Leibniz is reflected in the notation used today. Newton introduced the notation ${\displaystyle {\dot {f}}}$ for the derivative of a function f. Leibniz introduced the symbol ${\displaystyle \int }$ for the integral and wrote the derivative of a function y of the variable x as ${\displaystyle {\frac {dy}{dx}}}$, both of which are still in use.

Integrals

Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani's theorem (1821), David Bierens de Haan's work on the theory (1862) and his elaborate tables (1867), Lejeune Dirichlet's lectures (1858) embodied in Meyer's treatise (1871), and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlömilch, Elliott, Leudesdorf, and Kronecker are among the noteworthy contributions.

Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:

${\displaystyle \int _{0}^{1}x^{n-1}(1-x)^{n-1}\,dx}$

${\displaystyle \int _{0}^{\infty }e^{-x}x^{n-1}\,dx}$

although these were not the exact forms of Euler's study.

If n is a positive integer, it follows that:

${\displaystyle \int _{0}^{\infty }e^{-x}x^{n-1}dx=(n-1)!,}$

but the integral converges for all positive real ${\displaystyle n}$ and defines an analytic continuation of the factorial function to all of the complex plane except for poles at zero and the negative integers. To it Legendre assigned the symbol ${\displaystyle \Gamma }$, and it is now called the gamma function. Besides being analytic over positive reals ℝ⁺,  ${\displaystyle \Gamma }$ also enjoys the uniquely defining property that ${\displaystyle \log \Gamma }$ is convex, which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject Lejeune Dirichlet has contributed an important theorem (Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others. On the evaluation of ${\displaystyle \Gamma (x)}$ and ${\displaystyle \log \Gamma (x)}$ Raabe (1843–44), Bauer (1859), and Gudermann (1845) have written. Legendre's great table appeared in 1816.

Operational methods

Arbogast (1800) was the first to separate the symbol of operation from that of quantity in a differential equation. François (1812) and Servois (1814) seem to have been the first to give correct rules on the subject. Hargreave (1848) applied these methods in his memoir on differential equations, and Boole freely employed them. Grassmann and Hermann Hankel made great use of the theory, the former in studying equations, the latter in his theory of complex numbers.

Calculus of variations

The calculus of variations may be said to begin with a problem of Johann Bernoulli's (1696). It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Euler first elaborated the subject. His contributions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Lagrange contributed extensively to the theory, and Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810), Gauss (1829), Poisson (1831), Ostrogradsky (1834), and Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), Hesse (1857), Clebsch (1858), and Carll (1885), but perhaps the most important work of the century^{[peacock term]} is that of Weierstrass. His course on the theory may be asserted to be the first to place calculus on a firm and rigorous foundation.

Applications

The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the 18th century these applications were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis. To Lagrange (1773) we owe the introduction of the theory of the potential into dynamics, although the name "potential function" and the fundamental memoir of the subject are due to Green (1827, printed in 1828). The name "potential" is due to Gauss (1840), and the distinction between potential and potential function to Clausius. With its development are connected the names of Lejeune Dirichlet, Riemann, von Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami, and many of the leading physicists of the century.

It is impossible in this place to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations of Euler on vibrating chords; Sophie Germain on elastic membranes; Poisson, Lamé, Saint-Venant, and Clebsch on the elasticity of three-dimensional bodies; Fourier on heat diffusion; Fresnel on light; Maxwell, Helmholtz, and Hertz on electricity; Hansen, Hill, and Gyldén on astronomy; Maxwell on spherical harmonics; Lord Rayleigh on acoustics; and the contributions of Lejeune Dirichlet, Weber, Kirchhoff, F. Neumann, Lord Kelvin, Clausius, Bjerknes, MacCullagh, and Fuhrmann to physics in general. The labors of Helmholtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics.

Furthermore, infinitesimal calculus was introduced into the social sciences, starting with Neoclassical economics. Today, it is a valuable tool in mainstream economics.