Neurodegenerative disease

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Neurodegenerative_disease

 

Neurodegenerative disease
Normal brain on left contrasted with structural changes shown in brain on right of person with Alzheimer's disease, the most common neurodegenerative disease
Specialty	Neurology, Psychiatry

A neurodegenerative disease is caused by the progressive loss of neurons, in the process known as neurodegeneration. Neuronal damage may also ultimately result in their death. Neurodegenerative diseases include amyotrophic lateral sclerosis, multiple sclerosis, Parkinson's disease, Alzheimer's disease, Huntington's disease, multiple system atrophy, tauopathies, and prion diseases. Neurodegeneration can be found in the brain at many different levels of neuronal circuitry, ranging from molecular to systemic. Because there is no known way to reverse the progressive degeneration of neurons, these diseases are considered to be incurable; however research has shown that the two major contributing factors to neurodegeneration are oxidative stress and inflammation. Biomedical research has revealed many similarities between these diseases at the subcellular level, including atypical protein assemblies (like proteinopathy) and induced cell death. These similarities suggest that therapeutic advances against one neurodegenerative disease might ameliorate other diseases as well.

Within neurodegenerative diseases, it is estimated that 55 million people worldwide had dementia in 2019, and that by 2050 this figure will increase to 139 million people.

Specific disorders

The consequences of neurodegeneration can vary widely depending on the specific region affected, ranging from issues related to movement to the development of dementia.

Alzheimer's disease

Comparison of brain tissue between healthy individual and Alzheimer's disease patient, demonstrating extent of neuronal death

Main article: Alzheimer's disease

Alzheimer's disease (AD) is a chronic neurodegenerative disease that results in the loss of neurons and synapses in the cerebral cortex and certain subcortical structures, resulting in gross atrophy of the temporal lobe, parietal lobe, and parts of the frontal cortex and cingulate gyrus. It is the most common neurodegenerative disease. Even with billions of dollars being used to find a treatment for Alzheimer's disease, no effective treatments have been found. Within clinical trials stable and effective AD therapeutic strategies have a 99.5% failure rate. Reasons for this failure rate include inappropriate drug doses, invalid target and participant selection, and inadequate knowledge of pathophysiology of AD. Currently, diagnoses of Alzheimer's is subpar, and better methods need to be utilized for various aspects of clinical diagnoses. Alzheimer's has a 20% misdiagnosis rate.

AD pathology is primarily characterized by the presence of amyloid plaques and neurofibrillary tangles. Plaques are made up of small peptides, typically 39–43 amino acids in length, called amyloid beta (also written as A-beta or Aβ). Amyloid beta is a fragment from a larger protein called amyloid precursor protein (APP), a transmembrane protein that penetrates through the neuron's membrane. APP appears to play roles in normal neuron growth, survival and post-injury repair. APP is cleaved into smaller fragments by enzymes such as gamma secretase and beta secretase. One of these fragments gives rise to fibrils of amyloid beta which can self-assemble into the dense extracellular amyloid plaques.

Parkinson's disease

Main article: Parkinson's disease

Parkinson's disease (PD) is the second most common neurodegenerative disorder. It typically manifests as bradykinesia, rigidity, resting tremor and posture instability. The crude prevalence rate of PD has been reported to range from 15 per 100,000 to 12,500 per 100,000, and the incidence of PD from 15 per 100,000 to 328 per 100,000, with the disease being less common in Asian countries.

PD is primarily characterized by death of dopaminergic neurons in the substantia nigra, a region of the midbrain. The cause of this selective cell death is unknown. Notably, alpha-synuclein-ubiquitin complexes and aggregates are observed to accumulate in Lewy bodies within affected neurons. It is thought that defects in protein transport machinery and regulation, such as RAB1, may play a role in this disease mechanism. Impaired axonal transport of alpha-synuclein may also lead to its accumulation in Lewy bodies. Experiments have revealed reduced transport rates of both wild-type and two familial Parkinson's disease-associated mutant alpha-synucleins through axons of cultured neurons. Membrane damage by alpha-synuclein could be another Parkinson's disease mechanism.

The main known risk factor is age. Mutations in genes such as α-synuclein (SNCA), leucine-rich repeat kinase 2 (LRRK2), glucocerebrosidase (GBA), and tau protein (MAPT) can also cause hereditary PD or increase PD risk. While PD is the second most common neurodegenerative disorder, problems with diagnoses still persist. Problems with the sense of smell is a widespread symptom of Parkinson's disease (PD), however, some neurologists question its efficacy as a diagnostic tool. This assessment method is a source of controversy among medical professionals. The gut microbiome might play a role in the diagnosis of PD, and research suggests various ways that could revolutionize the future of PD treatment.

Huntington's disease

Main article: Huntington's disease

Huntington's disease (HD) is a rare autosomal dominant neurodegenerative disorder caused by mutations in the huntingtin gene (HTT). HD is characterized by loss of medium spiny neurons and astrogliosis. The first brain region to be substantially affected is the striatum, followed by degeneration of the frontal and temporal cortices. The striatum's subthalamic nuclei send control signals to the globus pallidus, which initiates and modulates motion. The weaker signals from subthalamic nuclei thus cause reduced initiation and modulation of movement, resulting in the characteristic movements of the disorder, notably chorea. Huntington's disease presents itself later in life even though the proteins that cause the disease works towards manifestation from their early stages in the humans affected by the proteins. Along with being a neurodegenerative disorder, HD has links to problems with neurodevelopment.

HD is caused by polyglutamine tract expansion in the huntingtin gene, resulting in the mutant huntingtin. Aggregates of mutant huntingtin form as inclusion bodies in neurons, and may be directly toxic. Additionally, they may damage molecular motors and microtubules to interfere with normal axonal transport, leading to impaired transport of important cargoes such as BDNF. Huntington's disease currently has no effective treatments that would modify the disease.

Multiple sclerosis

Main article: Multiple sclerosis

Multiple sclerosis (MS) is a chronic debilitating demyelinating disease of the central nervous system, caused by an autoimmune attack resulting in the progressive loss of myelin sheath on neuronal axons. The resultant decrease in the speed of signal transduction leads to a loss of functionality that includes both cognitive and motor impairment depending on the location of the lesion. The progression of MS occurs due to episodes of increasing inflammation, which is proposed to be due to the release of antigens such as myelin oligodendrocyte glycoprotein, myelin basic protein, and proteolipid protein, causing an autoimmune response. This sets off a cascade of signaling molecules that result in T cells, B cells, and macrophages crossing the blood-brain barrier and attacking myelin on neuronal axons leading to inflammation. Further release of antigens drives subsequent degeneration causing increased inflammation. Multiple sclerosis presents itself as a spectrum based on the degree of inflammation, a majority of patients experience early relapsing and remitting episodes of neuronal deterioration following a period of recovery. Some of these individuals may transition to a more linear progression of the disease, while about 15% of others begin with a progressive course on the onset of multiple sclerosis. The inflammatory response contributes to the loss of the grey matter, and as a result current literature devotes itself to combatting the auto-inflammatory aspect of the disease. While there are several proposed causal links between EBV and the HLA-DRB1*15:01 allele to the onset of MS – they may contribute to the degree of autoimmune attack and the resultant inflammation – they do not determine the onset of MS.

Amyotrophic lateral sclerosis

Main article: Amyotrophic lateral sclerosis

Amyotrophic lateral sclerosis (ALS), commonly referred to Lou Gehrig's disease, is a rare neurodegenerative disorder characterized by the gradual loss of both upper motor neurons (UMNs) and lower motor neurons (LMNs). Although initial symptoms may vary, most patients develop skeletal muscle weakness that progresses to involve the entire body. The precise etiology of ALS remains unknown. In 1993, missense mutations in the gene encoding the antioxidant enzyme superoxide dismutase 1 (SOD1) were discovered in a subset of patients with familial ALS. More recently, TAR DNA-binding protein 43 (TDP-43) and Fused in Sarcoma (FUS) protein aggregates have been implicated in some cases of the disease, and a mutation in chromosome 9 (C9orf72) is thought to be the most common known cause of sporadic ALS. Early diagnosis of ALS is harder than with other neurodegenerative diseases as there are no highly effective means of determining its early onset. Currently, there is research being done regarding the diagnosis of ALS through upper motor neuron tests. The Penn Upper Motor Neuron Score (PUMNS) consists of 28 criteria with a score range of 0–32. A higher score indicates a higher level of burden present on the upper motor neurons. The PUMNS has proven quite effective in determining the burden that exists on upper motor neurons in affected patients.

Independent research provided in vitro evidence that the primary cellular sites where SOD1 mutations act are located on astrocytes. Astrocytes then cause the toxic effects on the motor neurons. The specific mechanism of toxicity still needs to be investigated, but the findings are significant because they implicate cells other than neuron cells in neurodegeneration.

Batten disease

Main article: Batten disease

Batten disease is a rare and fatal recessive neurodegenerative disorder that begins in childhood. Batten disease is the common name for a group of lysosomal storage disorders known as neuronal ceroid lipofuscinoses (NCLs) – each caused by a specific gene mutation, of which there are thirteen. Since Batten disease is quite rare, its worldwide prevalence is about 1 in every 100,000 live births. In North America, NCL3 disease (juvenile NCL) typically manifests between the ages of 4 and 7. Batten disease is characterized by motor impairment, epilepsy, dementia, vision loss, and shortened lifespan. A loss of vision is common first sign of Batten disease. Loss of vision is typically preceded by cognitive and behavioral changes, seizures, and loss of the ability to walk. It is common for people to establish cardiac arrhythmias and difficulties eating food as the disease progresses. Batten disease diagnosis depends on a conflation of many criteria: clinical signs and symptoms, evaluations of the eye, electroencephalograms (EEG), and brain magnetic resonance imaging (MRI) results. The diagnosis provided by these results are corroborated by genetic and biochemical testing. It is only in recent years that more models have been created to expedite the research process for methods to treat Batten disease.

Creutzfeldt–Jakob disease

Main article: Creutzfeldt-Jakob disease

Creutzfeldt–Jakob disease (CJD) is a prion disease that is characterized by rapidly progressive dementia. Misfolded proteins called prions aggregate in brain tissue leading to nerve cell death. Variant Creutzfeldt–Jakob disease (vCJD) is the infectious form that comes from the meat of a cow that was infected with bovine spongiform encephalopathy, also called mad cow disease.

Risk factors

Aging

Further information: Aging brain

The greatest risk factor for neurodegenerative diseases is aging. Mitochondrial DNA mutations as well as oxidative stress both contribute to aging. Many of these diseases are late-onset, meaning there is some factor that changes as a person ages for each disease. One constant factor is that in each disease, neurons gradually lose function as the disease progresses with age. It has been proposed that DNA damage accumulation provides the underlying causative link between aging and neurodegenerative disease. About 20–40% of healthy people between 60 and 78 years old experience discernable decrements in cognitive performance in several domains including working, spatial, and episodic memory, and processing speed.

Infections

Risks from viral exposures according to one biobank study

A study using electronic health records indicates that 45 (with 22 of these being replicated with the UK Biobank) viral exposures can significantly elevate risks of neurodegenerative disease, including up to 15 years after infection.

Mechanisms

Genetics

See also: Trinucleotide repeat disorder and Epigenetics of neurodegenerative diseases

Many neurodegenerative diseases are caused by genetic mutations, most of which are located in completely unrelated genes. In many of the different diseases, the mutated gene has a common feature: a repeat of the CAG nucleotide triplet. CAG codes for the amino acid glutamine. A repeat of CAG results in a polyglutamine (polyQ) tract. Diseases associated with such mutations are known as trinucleotide repeat disorders.

Polyglutamine repeats typically cause dominant pathogenesis. Extra glutamine residues can acquire toxic properties through a variety of ways, including irregular protein folding and degradation pathways, altered subcellular localization, and abnormal interactions with other cellular proteins. PolyQ studies often use a variety of animal models because there is such a clearly defined trigger – repeat expansion. Extensive research has been done using the models of nematode (C. elegans), and fruit fly (Drosophila), mice, and non-human primates.

Nine inherited neurodegenerative diseases are caused by the expansion of the CAG trinucleotide and polyQ tract, including Huntington's disease and the spinocerebellar ataxias.

Epigenetics

The presence of epigenetic modifications for certain genes has been demonstrated in this type of pathology. An example is FKBP5 gene, which progressively increases its expression with age and has been related to Braak staging and increased tau pathology both in vitro and in mouse models of AD.

Protein misfolding

See also: Stress granule

Several neurodegenerative diseases are classified as proteopathies as they are associated with the aggregation of misfolded proteins. Protein toxicity is one of the key mechanisms of many neurodegenrative diseases.

• alpha-synuclein: can aggregate to form insoluble fibrils in pathological conditions characterized by Lewy bodies, such as Parkinson's disease, dementia with Lewy bodies, and multiple system atrophy. Alpha-synuclein is the primary structural component of Lewy body fibrils. In addition, an alpha-synuclein fragment, known as the non-Abeta component (NAC), is found in amyloid plaques in Alzheimer's disease.
• tau: hyperphosphorylated tau protein is the main component of neurofibrillary tangles in Alzheimer's disease; tau fibrils are the main component of Pick bodies found in behavioral variant frontotemporal dementia.
• amyloid beta: the major component of amyloid plaques in Alzheimer's disease.
• prion: main component of prion diseases and transmissible spongiform encephalopathy.

Intracellular mechanisms

Protein degradation pathways

Parkinson's disease and Huntington's disease are both late-onset and associated with the accumulation of intracellular toxic proteins. Diseases caused by the aggregation of proteins are known as proteopathies, and they are primarily caused by aggregates in the following structures:

• cytosol, e.g. Parkinson's and Huntington's
• nucleus, e.g. Spinocerebellar ataxia type 1 (SCA1)
• endoplasmic reticulum (ER), (as seen with neuroserpin mutations that cause familial encephalopathy with neuroserpin inclusion bodies)
• extracellularly excreted proteins, amyloid-beta in Alzheimer's disease

There are two main avenues eukaryotic cells use to remove troublesome proteins or organelles:

• ubiquitin–proteasome: protein ubiquitin along with enzymes is key for the degradation of many proteins that cause proteopathies including polyQ expansions and alpha-synucleins. Research indicates proteasome enzymes may not be able to correctly cleave these irregular proteins, which could possibly result in a more toxic species. This is the primary route cells use to degrade proteins.
  • Decreased proteasome activity is consistent with models in which intracellular protein aggregates form. It is still unknown whether or not these aggregates are a cause or a result of neurodegeneration.
• autophagy–lysosome pathways: a form of programmed cell death (PCD), this becomes the favorable route when a protein is aggregate-prone meaning it is a poor proteasome substrate. This can be split into two forms of autophagy: macroautophagy and chaperone-mediated autophagy (CMA).
  • macroautophagy is involved with nutrient recycling of macromolecules under conditions of starvation, certain apoptotic pathways, and if absent, leads to the formation of ubiquinated inclusions. Experiments in mice with neuronally confined macroautophagy-gene knockouts develop intraneuronal aggregates leading to neurodegeneration.
  • chaperone-mediated autophagy defects may also lead to neurodegeneration. Research has shown that mutant proteins bind to the CMA-pathway receptors on lysosomal membrane and in doing so block their own degradation as well as the degradation of other substrates.

Membrane damage

Damage to the membranes of organelles by monomeric or oligomeric proteins could also contribute to these diseases. Alpha-synuclein can damage membranes by inducing membrane curvature, and cause extensive tubulation and vesiculation when incubated with artificial phospholipid vesicles. In addition, oligomeric α-synuclein species can form nanoscale pores in lipid membranes, further contributing to membrane disruption. The tubes formed from these lipid vesicles consist of both micellar as well as bilayer tubes. Extensive induction of membrane curvature is deleterious to the cell and would eventually lead to cell death. Apart from tubular structures, alpha-synuclein can also form lipoprotein nanoparticles similar to apolipoproteins.

Mitochondrial dysfunction

The most common form of cell death in neurodegeneration is through the intrinsic mitochondrial apoptotic pathway. This pathway controls the activation of caspase-9 by regulating the release of cytochrome c from the mitochondrial intermembrane space. Reactive oxygen species (ROS) are normal byproducts of mitochondrial respiratory chain activity. ROS concentration is mediated by mitochondrial antioxidants such as manganese superoxide dismutase (SOD2) and glutathione peroxidase. Over production of ROS (oxidative stress) is a central feature of all neurodegenerative disorders. In addition to the generation of ROS, mitochondria are also involved with life-sustaining functions including calcium homeostasis, PCD, mitochondrial fission and fusion, lipid concentration of the mitochondrial membranes, and the mitochondrial permeability transition. Mitochondrial disease leading to neurodegeneration is likely, at least on some level, to involve all of these functions.

There is strong evidence that mitochondrial dysfunction and oxidative stress play a causal role in neurodegenerative disease pathogenesis, including in four of the more well known diseases Alzheimer's, Parkinson's, Huntington's, and amyotrophic lateral sclerosis.

Neurons are particularly vulnerable to oxidative damage due to their strong metabolic activity associated with high transcription levels, high oxygen consumption, and weak antioxidant defense.

DNA damage

The brain metabolizes as much as a fifth of consumed oxygen, and reactive oxygen species produced by oxidative metabolism are a major source of DNA damage in the brain. Damage to a cell's DNA is particularly harmful because DNA is the blueprint for protein production and unlike other molecules it cannot simply be replaced by re-synthesis. The vulnerability of post-mitotic neurons to DNA damage (such as oxidative lesions or certain types of DNA strand breaks), coupled with a gradual decline in the activities of repair mechanisms, could lead to accumulation of DNA damage with age and contribute to brain aging and neurodegeneration. DNA single-strand breaks are common and are associated with the neurodegenerative disease ataxia-oculomotor apraxia. Increased oxidative DNA damage in the brain is associated with Alzheimer's disease and Parkinson's disease. Defective DNA repair has been linked to neurodegenerative disorders such as Alzheimer's disease, amyotrophic lateral sclerosis, ataxia telangiectasia, Cockayne syndrome, Parkinson's disease and xeroderma pigmentosum.

Axonal transport

Main article: Axonal transport

Axonal swelling, and axonal spheroids have been observed in many different neurodegenerative diseases. This suggests that defective axons are not only present in diseased neurons, but also that they may cause certain pathological insult due to accumulation of organelles. Axonal transport can be disrupted by a variety of mechanisms including damage to: kinesin and cytoplasmic dynein, microtubules, cargoes, and mitochondria. When axonal transport is severely disrupted a degenerative pathway known as Wallerian-like degeneration is often triggered.

Programmed cell death

Programmed cell death (PCD) is death of a cell in any form, mediated by an intracellular program. This process can be activated in neurodegenerative diseases including Parkinson's disease, amytrophic lateral sclerosis, Alzheimer's disease and Huntington's disease. PCD observed in neurodegenerative diseases may be directly pathogenic; alternatively, PCD may occur in response to other injury or disease processes.

Apoptosis (type I)

Apoptosis is a form of programmed cell death in multicellular organisms. It is one of the main types of programmed cell death (PCD) and involves a series of biochemical events leading to a characteristic cell morphology and death.

• Extrinsic apoptotic pathways: Occur when factors outside the cell activate cell surface death receptors (e.g., Fas) that result in the activation of caspases-8 or -10.
• Intrinsic apoptotic pathways: Result from mitochondrial release of cytochrome c or endoplasmic reticulum malfunctions, each leading to the activation of caspase-9. The nucleus and Golgi apparatus are other organelles that have damage sensors, which can lead the cells down apoptotic pathways.

Caspases (cysteine-aspartic acid proteases) cleave at very specific amino acid residues. There are two types of caspases: initiators and effectors. Initiator caspases cleave inactive forms of effector caspases. This activates the effectors that in turn cleave other proteins resulting in apoptotic initiation.

Autophagic (type II)

Autophagy is a form of intracellular phagocytosis in which a cell actively consumes damaged organelles or misfolded proteins by encapsulating them into an autophagosome, which fuses with a lysosome to destroy the contents of the autophagosome. Because many neurodegenerative diseases show unusual protein aggregates, it is hypothesized that defects in autophagy could be a common mechanism of neurodegeneration.

Cytoplasmic (type III)

PCD can also occur via non-apoptotic processes, also known as Type III or cytoplasmic cell death. For example, type III PCD might be caused by trophotoxicity, or hyperactivation of trophic factor receptors. Cytotoxins that induce PCD can cause necrosis at low concentrations, or aponecrosis (combination of apoptosis and necrosis) at higher concentrations. It is still unclear exactly what combination of apoptosis, non-apoptosis, and necrosis causes different kinds of aponecrosis.

Transglutaminase

Transglutaminases are human enzymes ubiquitously present in the human body and in the brain in particular.

The main function of transglutaminases is bind proteins and peptides intra- and intermolecularly, by a type of covalent bonds termed isopeptide bonds, in a reaction termed transamidation or crosslinking.

Transglutaminase binding of these proteins and peptides make them clump together. The resulting structures are turned extremely resistant to chemical and mechanical disruption.

Most relevant human neurodegenerative diseases share the property of having abnormal structures made up of proteins and peptides.

Each of these neurodegenerative diseases have one (or several) specific main protein or peptide. In Alzheimer's disease, these are amyloid-beta and tau. In Parkinson's disease, it is alpha-synuclein. In Huntington's disease, it is huntingtin.

Transglutaminase substrates: Amyloid-beta, tau, alpha-synuclein and huntingtin have been proved to be substrates of transglutaminases in vitro or in vivo, that is, they can be bonded by trasglutaminases by covalent bonds to each other and potentially to any other transglutaminase substrate in the brain.

Transglutaminase augmented expression: It has been proved that in these neurodegenerative diseases (Alzheimer's disease, Parkinson's disease, and Huntington's disease) the expression of the transglutaminase enzyme is increased.

Presence of isopeptide bonds in these structures: The presence of isopeptide bonds (the result of the transglutaminase reaction) have been detected in the abnormal structures that are characteristic of these neurodegenerative diseases.

Co-localization: Co-localization of transglutaminase mediated isopeptide bonds with these abnormal structures has been detected in the autopsy of brains of patients with these diseases.

Management

The process of neurodegeneration is not well understood, so the diseases that stem from it have, as yet, no cures.

Animal models in research

In the search for effective treatments (as opposed to palliative care), investigators employ animal models of disease to test potential therapeutic agents. Model organisms provide an inexpensive and relatively quick means to perform two main functions: target identification and target validation. Together, these help show the value of any specific therapeutic strategies and drugs when attempting to ameliorate disease severity. An example is the drug Dimebon by Medivation, Inc. In 2009 this drug was in phase III clinical trials for use in Alzheimer's disease, and also phase II clinical trials for use in Huntington's disease. In March 2010, the results of a clinical trial phase III were released; the investigational Alzheimer's disease drug Dimebon failed in the pivotal CONNECTION trial of patients with mild-to-moderate disease. With CONCERT, the remaining Pfizer and Medivation Phase III trial for Dimebon (latrepirdine) in Alzheimer's disease failed in 2012, effectively ending the development in this indication.

In another experiment using a rat model of Alzheimer's disease, it was demonstrated that systemic administration of hypothalamic proline-rich peptide (PRP)-1 offers neuroprotective effects and can prevent neurodegeneration in hippocampus amyloid-beta 25–35. This suggests that there could be therapeutic value to PRP-1.

Other avenues of investigation

Protein degradation offers therapeutic options both in preventing the synthesis and degradation of irregular proteins. There is also interest in upregulating autophagy to help clear protein aggregates implicated in neurodegeneration. Both of these options involve very complex pathways that we are only beginning to understand.

The goal of immunotherapy is to enhance aspects of the immune system. Both active and passive vaccinations have been proposed for Alzheimer's disease and other conditions; however, more research must be done to prove safety and efficacy in humans.

A current therapeutic target for the treatment of Alzheimer's disease is the protease β-secretase, which is involved in the amyloidogenic processing pathway that leads to the pathological accumulation of proteins in the brain. When the gene that encodes for amyloid precursor protein (APP) is spliced by α-secretase rather than β-secretase, the toxic protein β amyloid is not produced. Targeted inhibition of β-secretase can potentially prevent the neuronal death that is responsible for the symptoms of Alzheimer's disease.

Derivative

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Derivative

 

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

 

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

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. 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. 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. The process of finding a derivative is called differentiation.

There are multiple different notations for differentiation. Leibniz notation, named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz notation by adding superscripts to the differentials, and in prime notation by adding additional prime marks. Higher order derivatives are used in physics; for example, the first derivative with respect to time of the position of a moving object is its velocity, and the second derivative is its acceleration.

Derivatives can be generalized to functions of several real variables. In this case, 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.

Definition

As a limit

A function of a real variable ${\displaystyle f(x)}$ is differentiable at a point ${\displaystyle a}$ of its domain, if its domain contains an open interval containing ⁠${\displaystyle a}$⁠, and the limit $${\displaystyle L=\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}}$$ exists. This means that, for every positive real number ⁠${\displaystyle \epsilon }$⁠, there exists a positive real number ${\displaystyle \delta }$ such that, for every ${\displaystyle h}$ such that ${\displaystyle |h|<\delta }$ and ${\displaystyle h\ne 0}$ then ${\displaystyle f(a+h)}$ is defined, and $${\displaystyle \left|L-\frac{f(a+h)-f(a)}{h}\right|<\epsilon ,}$$ where the vertical bars denote the absolute value. This is an example of the (ε, δ)-definition of limit.

If the function ${\displaystyle f}$ is differentiable at ⁠${\displaystyle a}$⁠, that is if the limit ${\displaystyle L}$ exists, then this limit is called the derivative of ${\displaystyle f}$ at ⁠${\displaystyle a}$⁠. Multiple notations for the derivative exist. The derivative of ${\displaystyle f}$ at ${\displaystyle a}$ can be denoted ⁠${\displaystyle f^{\prime }(a)}$⁠, read as "⁠${\displaystyle f}$⁠ prime of ⁠${\displaystyle a}$⁠"; or it can be denoted ⁠${\displaystyle \frac{df}{dx}(a)}$⁠, read as "the derivative of ${\displaystyle f}$ with respect to ${\displaystyle x}$ at ⁠${\displaystyle a}$⁠" or "⁠${\displaystyle df}$⁠ by (or over) ${\displaystyle dx}$ at ⁠${\displaystyle a}$⁠". See § Notation below. If ${\displaystyle f}$ is a function that has a derivative at every point in its domain, then a function can be defined by mapping every point ${\displaystyle x}$ to the value of the derivative of ${\displaystyle f}$ at ⁠${\displaystyle x}$⁠. This function is written ${\displaystyle f^{\prime }}$ and is called the derivative function or the derivative of ⁠${\displaystyle f}$⁠. The function ${\displaystyle f}$ sometimes has a derivative at most, but not all, points of its domain. The function whose value at ${\displaystyle a}$ equals ${\displaystyle f^{\prime }(a)}$ whenever ${\displaystyle f^{\prime }(a)}$ is defined and elsewhere is undefined is also called the derivative of ⁠${\displaystyle f}$⁠. It is still a function, but its domain may be smaller than the domain of ⁠${\displaystyle f}$⁠.

For example, let ${\displaystyle f}$ be the squaring function: ⁠${\displaystyle f(x)=x^2}$⁠. Then the quotient in the definition of the derivative is $${\displaystyle \frac{f(a+h)-f(a)}{h}=\frac{(a+h{)}^2-a^2}{h}=\frac{a^2+2ah+h^2-a^2}{h}=2a+h.}$$ The division in the last step is valid as long as ⁠${\displaystyle h\ne 0}$⁠. The closer ${\displaystyle h}$ is to ⁠${\displaystyle 0}$⁠, the closer this expression becomes to the value ⁠${\displaystyle 2a}$⁠. The limit exists, and for every input ${\displaystyle a}$ the limit is ⁠${\displaystyle 2a}$⁠. So, the derivative of the squaring function is the doubling function: ⁠${\displaystyle f^{\prime }(x)=2x}$⁠.

The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function ⁠${\displaystyle f}$⁠, specifically the points ${\displaystyle (a,f(a))}$ and ⁠${\displaystyle (a+h,f(a+h))}$⁠. As ${\displaystyle h}$ is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the tangent to the graph of ${\displaystyle f}$ at ⁠${\displaystyle a}$⁠. In other words, the derivative is the slope of the tangent.

Using infinitesimals

One way to think of the derivative $\frac{df}{dx}(a)$ is as the ratio of an infinitesimal change in the output of the function ${\displaystyle f}$ to an infinitesimal change in its input. In order to make this intuition rigorous, a system of rules for manipulating infinitesimal quantities is required. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals are an extension of the real numbers that contain numbers greater than anything of the form ${\displaystyle 1+1+\cdots +1}$ for any finite number of terms. Such numbers are infinite, and their reciprocals are infinitesimals. The application of hyperreal numbers to the foundations of calculus is called nonstandard analysis. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the ${\displaystyle d}$ in the Leibniz notation. Thus, the derivative of ${\displaystyle f(x)}$ becomes $${\displaystyle f^{\prime }(x)=\mathrm{st}\left(\frac{f(x+dx)-f(x)}{dx}\right)}$$ for an arbitrary infinitesimal ⁠${\displaystyle dx}$⁠, where ${\displaystyle \mathrm{st}}$ denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Taking the squaring function ${\displaystyle f(x)=x^2}$ as an example again, $${\displaystyle \begin{array}{rl}{f}^{\prime }(x)& =\mathrm{st}\left(\frac{x^2+2x\cdot dx+(dx{)}^2-x^2}{dx}\right)\\ & =\mathrm{st}\left(\frac{2x\cdot dx+(dx{)}^2}{dx}\right)\\ & =\mathrm{st}\left(\frac{2x\cdot dx}{dx}+\frac{(dx{)}^2}{dx}\right)\\ & =\mathrm{st}\left(2x+dx\right)\\ & =2x.\end{array}}$$

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).

 

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.

If ${\displaystyle f}$ is differentiable at ⁠${\displaystyle a}$⁠, then ${\displaystyle f}$ must also be continuous at ⁠${\displaystyle a}$⁠. As an example, choose a point ${\displaystyle a}$ and let ${\displaystyle f}$ be the step function that returns the value 1 for all ${\displaystyle x}$ less than ⁠${\displaystyle a}$⁠, and returns a different value 10 for all ${\displaystyle x}$ greater than or equal to ⁠${\displaystyle a}$⁠. The function ${\displaystyle f}$ cannot have a derivative at ⁠${\displaystyle a}$⁠. If ${\displaystyle h}$ is negative, then ${\displaystyle a+h}$ is on the low part of the step, so the secant line from ${\displaystyle a}$ to ${\displaystyle a+h}$ is very steep; as ${\displaystyle h}$ tends to zero, the slope tends to infinity. If ${\displaystyle h}$ is positive, then ${\displaystyle a+h}$ is on the high part of the step, so the secant line from ${\displaystyle a}$ to ${\displaystyle a+h}$ has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by ${\displaystyle f(x)=|x|}$ is continuous at ⁠${\displaystyle x=0}$⁠, but it is not differentiable there. If ${\displaystyle h}$ is positive, then the slope of the secant line from 0 to ${\displaystyle h}$ is one; if ${\displaystyle h}$ is negative, then the slope of the secant line from ${\displaystyle 0}$ to ${\displaystyle h}$ is ⁠${\displaystyle -1}$⁠. This can be seen graphically as a "kink" or a "cusp" in the graph at ⁠${\displaystyle x=0}$⁠. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by ${\displaystyle f(x)=x^{1/3}}$ is not differentiable at ⁠${\displaystyle x=0}$⁠. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.

Most functions that occur in practice have derivatives at all points or 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 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 any random continuous functions have a derivative at even one point.

Notation

Main article: Notation for differentiation

One common way of writing the derivative of a function is Leibniz notation, introduced by Gottfried Wilhelm Leibniz in 1675, which denotes a derivative as the quotient of two differentials, such as ${\displaystyle dy}$ and ⁠${\displaystyle dx}$⁠. It is still commonly used when the equation ${\displaystyle y=f(x)}$ is viewed as a functional relationship between dependent and independent variables. The first derivative is denoted by ⁠${\displaystyle \frac{dy}{dx}}$⁠, read as "the derivative of ${\displaystyle y}$ with respect to ⁠${\displaystyle x}$⁠". This derivative can alternately be treated as the application of a differential operator to a function, $\frac{dy}{dx}=\frac{d}{dx}f(x).$ Higher derivatives are expressed using the notation $\frac{d^{n}y}{d{x}^n}$ for the ${\displaystyle n}$th derivative of ⁠${\displaystyle y=f(x)}$⁠. These are abbreviations for multiple applications of the derivative operator; for example, $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}(\frac{d}{dx}f(x)).$ Unlike some alternatives, Leibniz notation involves explicit specification of the variable for differentiation, in the denominator, which removes ambiguity when working with multiple interrelated quantities. The derivative of a composed function can be expressed using the chain rule: if ${\displaystyle u=g(x)}$ and ${\displaystyle y=f(g(x))}$ then $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}.$

Another common notation for differentiation is by using the prime mark in the symbol of a function ⁠${\displaystyle f(x)}$⁠. This notation, due to Joseph-Louis Lagrange, is now known as prime notation. The first derivative is written as ⁠${\displaystyle f^{\prime }(x)}$⁠, read as "⁠${\displaystyle f}$⁠ prime of ⁠${\displaystyle x}$⁠", or ⁠${\displaystyle y^{\prime }}$⁠, read as "⁠${\displaystyle y}$⁠ prime". Similarly, the second and the third derivatives can be written as ${\displaystyle f^{″}}$ and ⁠${\displaystyle f^{‴}}$⁠, respectively. For denoting the number of higher derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses, such as ${\displaystyle f^{\mathrm{i}\mathrm{v}}}$ or ⁠${\displaystyle f^{(4)}}$⁠. The latter notation generalizes to yield the notation ${\displaystyle f^{(n)}}$ for the ⁠${\displaystyle n}$⁠th derivative of ⁠${\displaystyle f}$⁠.

In Newton's notation or the dot notation, a dot is placed over a symbol to represent a time derivative. If ${\displaystyle y}$ is a function of ⁠${\displaystyle t}$⁠, then the first and second derivatives can be written as ${\displaystyle \dot{y}}$ and ⁠${\displaystyle \ddot{y}}$⁠, respectively. This notation is used exclusively for derivatives with respect to time or arc length. It is typically used in differential equations in physics and differential geometry. However, the dot notation becomes unmanageable for high-order derivatives (of order 4 or more) and cannot deal with multiple independent variables.

Another notation is D-notation, which represents the differential operator by the symbol ⁠${\displaystyle D}$⁠. The first derivative is written ${\displaystyle Df(x)}$ and higher derivatives are written with a superscript, so the ${\displaystyle n}$th derivative is ⁠${\displaystyle D^{n}f(x)}$⁠. This notation is sometimes called Euler notation, although it seems that Leonhard Euler did not use it, and the notation was introduced by Louis François Antoine Arbogast. To indicate a partial derivative, the variable differentiated by is indicated with a subscript, for example given the function ⁠${\displaystyle u=f(x,y)}$⁠, its partial derivative with respect to ${\displaystyle x}$ can be written ${\displaystyle D_{x}u}$ or ⁠${\displaystyle D_{x}f(x,y)}$⁠. Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g. $D_{xy}f(x,y)=\frac{\mathrm{\partial }}{\mathrm{\partial }y}(\frac{\mathrm{\partial }}{\mathrm{\partial }x}f(x,y))$ and ⁠${\displaystyle D_x^{2}f(x,y)=\frac{\mathrm{\partial }}{\mathrm{\partial }x}(\frac{\mathrm{\partial }}{\mathrm{\partial }x}f(x,y))}$⁠.

Rules of computation

Main article: Differentiation rules

In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. 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. This process of finding a derivative is known as differentiation.

Rules for basic functions

The following are the rules for the derivatives of the most common basic functions. Here, ${\displaystyle a}$ is a real number, and ${\displaystyle e}$ is the base of the natural logarithm, approximately 2.71828.

• Derivatives of powers:

  ${\displaystyle \frac{d}{dx}{x}^a=a{x}^{a-1}}$

• Functions of exponential, natural logarithm, and logarithm with general base:

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

  ⁠${\displaystyle \frac{d}{dx}{a}^x=a^x\ln (a)}$⁠, for ${\displaystyle a>0}$

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

  ⁠${\displaystyle \frac{d}{dx}{\log }_a(x)=\frac{1}{x\ln (a)}}$⁠, for ${\displaystyle x,a>0}$

• 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:

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

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

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

Rules for combined functions

The following rules allow deducing derivatives of many functions from the derivatives of the basic functions:

• Constant rule: if ${\displaystyle f}$ is a constant function, then for all ⁠${\displaystyle x}$⁠,

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

• Sum rule:

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

• Product rule:

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

• Quotient rule:

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

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

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

Computation example

The derivative of the function given by ${\displaystyle f(x)=x^4+\sin \left(x^2\right)-\ln (x)e^x+7}$ is $${\displaystyle \begin{array}{rl}{f}^{\prime }(x)& =4x^{(4-1)}+\frac{d\left(x^2\right)}{dx}\cos \left(x^2\right)-\frac{d\left(\ln x\right)}{dx}{e}^x-\ln (x)\frac{d\left(e^x\right)}{dx}+0\\ & =4x^3+2x\cos \left(x^2\right)-\frac{1}{x}{e}^x-\ln (x)e^x.\end{array}}$$ Here the second term was computed using the chain rule and the third term using the product rule. The known derivatives of the elementary functions ⁠${\displaystyle x^2}$⁠, ⁠${\displaystyle x^4}$⁠, ⁠${\displaystyle \sin (x)}$⁠, ⁠${\displaystyle \ln (x)}$⁠, and ⁠${\displaystyle \exp (x)}$⁠, as well as the constant ⁠${\displaystyle 7}$⁠, were also used.

Antidifferentiation

Main article: Antiderivative

An antiderivative of a function ${\displaystyle f}$ is a function whose derivative is ⁠${\displaystyle f}$⁠. Antiderivatives are not unique: if ${\displaystyle A}$ is an antiderivative of ⁠${\displaystyle f}$⁠, then so is ⁠${\displaystyle A+c}$⁠, where ${\displaystyle c}$ is any constant, because the derivative of a constant is zero. The fundamental theorem of calculus shows that finding an antiderivative of a function gives a way to compute the areas of shapes bounded by that function. More precisely, the integral of a function over a closed interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of that interval.

Higher-order derivatives

Higher-order derivatives are the result of differentiating a function repeatedly. Given that ${\displaystyle f}$ is a differentiable function, the derivative of ${\displaystyle f}$ is the first derivative, denoted as ⁠${\displaystyle f^{\prime }}$⁠. The derivative of ${\displaystyle f^{\prime }}$ is the second derivative, denoted as ⁠${\displaystyle f^{″}}$⁠, and the derivative of ${\displaystyle f^{″}}$ is the third derivative, denoted as ⁠${\displaystyle f^{‴}}$⁠. By continuing this process, if it exists, the ⁠${\displaystyle n}$⁠th derivative is the derivative of the ⁠${\displaystyle (n-1)}$⁠th derivative or the derivative of order ⁠${\displaystyle n}$⁠. As has been discussed above, the generalization of derivative of a function ${\displaystyle f}$ may be denoted as ⁠${\displaystyle f^{(n)}}$⁠. A function that has ${\displaystyle k}$ successive derivatives is called ${\displaystyle k}$ times differentiable. If the ${\displaystyle k}$-th derivative is continuous, then the function is said to be of differentiability class ⁠${\displaystyle C^k}$⁠. A function that has infinitely many derivatives is called infinitely differentiable or smooth. Any polynomial function is infinitely differentiable; taking derivatives repeatedly will eventually result in a constant function, and all subsequent derivatives of that function are zero.

One application of higher-order derivatives is in physics. For example, if the function ⁠${\displaystyle x}$⁠ represents an object's position with respect to time, ⁠${\displaystyle x^{\prime }}$⁠ represents the object's velocity, ⁠${\displaystyle x^{″}}$⁠ represents the object's acceleration, and ⁠${\displaystyle x^{‴}}$⁠ represents the object's jerk.

In other dimensions

See also: Vector calculus and Multivariable calculus

Vector-valued functions

A vector-valued function ${\displaystyle \mathbf{y}}$ of a real variable sends real numbers to vectors in some vector space ⁠${\displaystyle {\mathbb{R}}^n}$⁠. A vector-valued function can be split up into its coordinate functions ⁠${\displaystyle y_1(t),y_2(t),\dots ,y_n(t)}$⁠, meaning that ⁠${\displaystyle 1}$⁠. This includes, for example, parametric curves in ${\displaystyle {\mathbb{R}}^2}$ or ⁠${\displaystyle {\mathbb{R}}^3}$⁠. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of ${\displaystyle \mathbf{y}(t)}$ is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is, $${\displaystyle {\mathbf{y}}^{\prime }(t)=\underset{h\to 0}{lim}\frac{\mathbf{y}(t+h)-\mathbf{y}(t)}{h},}$$ if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of ${\displaystyle \mathbf{y}}$ exists for every value of ⁠${\displaystyle t}$⁠, then ${\displaystyle {\mathbf{y}}^{\prime }}$ is another vector-valued function.

Partial derivatives

Main article: Partial derivative

Functions can depend upon more than one variable. A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry. As with ordinary derivatives, multiple notations exist: the partial derivative of a function ${\displaystyle f(x,y,\dots )}$ with respect to the variable ${\displaystyle x}$ is variously denoted by

⁠${\displaystyle f_x}$⁠, ⁠${\displaystyle f_x^{\prime }}$⁠, ⁠${\displaystyle {\mathrm{\partial }}_{x}f}$⁠, ⁠${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}f}$⁠, or ⁠${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}$⁠,

among other possibilities. It can be thought of as the rate of change of the function in the ${\displaystyle x}$-direction. 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". For example, let ⁠${\displaystyle f(x,y)=x^2+xy+y^2}$⁠, then the partial derivative of function ${\displaystyle f}$ with respect to both variables ${\displaystyle x}$ and ${\displaystyle y}$ are, respectively: $${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}=2x+y,\frac{\mathrm{\partial }f}{\mathrm{\partial }y}=x+2y.}$$ In general, the partial derivative of a function ${\displaystyle f(x_1,\dots ,x_n)}$ in the direction ${\displaystyle x_i}$ at the point ${\displaystyle (a_1,\dots ,a_n)}$ is defined to be: $${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}(a_1,\dots ,a_n)=\underset{h\to 0}{lim}\frac{f(a_1,\dots ,a_i+h,\dots ,a_n)-f(a_1,\dots ,a_i,\dots ,a_n)}{h}.}$$

This is fundamental for the study of the functions of several real variables. Let ${\displaystyle f(x_1,\dots ,x_n)}$ be such a real-valued function. If all partial derivatives ${\displaystyle f}$ with respect to ${\displaystyle x_j}$ are defined at the point ⁠${\displaystyle (a_1,\dots ,a_n)}$⁠, these partial derivatives define the vector $${\displaystyle \mathrm{\nabla }f(a_1,\dots ,a_n)=\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_1}(a_1,\dots ,a_n),\dots ,\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_n}(a_1,\dots ,a_n)\right),}$$ which is called the gradient of ${\displaystyle f}$ at ⁠${\displaystyle a}$⁠. If ${\displaystyle f}$ is differentiable at every point in some domain, then the gradient is a vector-valued function ${\displaystyle \mathrm{\nabla }f}$ that maps the point ${\displaystyle (a_1,\dots ,a_n)}$ to the vector ⁠${\displaystyle \mathrm{\nabla }f(a_1,\dots ,a_n)}$⁠. Consequently, the gradient determines a vector field.

Directional derivatives

Main article: Directional derivative

If ${\displaystyle f}$ is a real-valued function on ⁠${\displaystyle {\mathbb{R}}^n}$⁠, then the partial derivatives of ${\displaystyle f}$ measure its variation in the direction of the coordinate axes. For example, if ${\displaystyle f}$ is a function of ${\displaystyle x}$ and ⁠${\displaystyle y}$⁠, then its partial derivatives measure the variation in ${\displaystyle f}$ in the ${\displaystyle x}$ and ${\displaystyle y}$ direction. However, they do not directly measure the variation of ${\displaystyle f}$ in any other direction, such as along the diagonal line ⁠${\displaystyle y=x}$⁠. These are measured using directional derivatives. Given a vector ⁠${\displaystyle \mathbf{v}=(v_1,\dots ,v_n)}$⁠, then the directional derivative of ${\displaystyle f}$ in the direction of ${\displaystyle \mathbf{v}}$ at the point ${\displaystyle \mathbf{x}}$ is: $${\displaystyle D_{\mathbf{v}}f(\mathbf{x})=\underset{h\to 0}{lim}\frac{f(\mathbf{x}+h\mathbf{v})-f(\mathbf{x})}{h}.}$$

If all the partial derivatives of ${\displaystyle f}$ exist and are continuous at ⁠${\displaystyle \mathbf{x}}$⁠, then they determine the directional derivative of ${\displaystyle f}$ in the direction ${\displaystyle \mathbf{v}}$ by the formula: $${\displaystyle D_{\mathbf{v}}f(\mathbf{x})=\sum _{j=1}^n{v}_j\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_j}.}$$

Total derivative and Jacobian matrix

Main article: Total derivative

When ${\displaystyle f}$ is a function from an open subset of ${\displaystyle {\mathbb{R}}^n}$ to ⁠${\displaystyle {\mathbb{R}}^m}$⁠, then the directional derivative of ${\displaystyle f}$ in a chosen direction is the best linear approximation to ${\displaystyle f}$ at that point and in that direction. However, when ⁠${\displaystyle n>1}$⁠, no single directional derivative can give a complete picture of the behavior of ⁠${\displaystyle f}$⁠. The total derivative gives a complete picture by considering all directions at once. That is, for any vector ${\displaystyle \mathbf{v}}$ starting at ⁠${\displaystyle \mathbf{a}}$⁠, the linear approximation formula holds: $${\displaystyle f(\mathbf{a}+\mathbf{v})\approx f(\mathbf{a})+f^{\prime }(\mathbf{a})\mathbf{v}.}$$ Similarly with the single-variable derivative, ${\displaystyle f^{\prime }(\mathbf{a})}$ is chosen so that the error in this approximation is as small as possible. The total derivative of ${\displaystyle f}$ at ${\displaystyle \mathbf{a}}$ is the unique linear transformation ${\displaystyle f^{\prime }(\mathbf{a}):{\mathbb{R}}^n\to {\mathbb{R}}^m}$ such that $${\displaystyle \underset{\mathbf{h}\to 0}{lim}\frac{\Vert f(\mathbf{a}+\mathbf{h})-(f(\mathbf{a})+f^{\prime }(\mathbf{a})\mathbf{h})\Vert }{\Vert \mathbf{h}\Vert }=0.}$$ Here ${\displaystyle \mathbf{h}}$ is a vector in ⁠${\displaystyle {\mathbb{R}}^n}$⁠, so the norm in the denominator is the standard length on ⁠${\displaystyle {\mathbb{R}}^n}$⁠. However, ${\displaystyle f^{\prime }(\mathbf{a})\mathbf{h}}$ is a vector in ⁠${\displaystyle {\mathbb{R}}^m}$⁠, and the norm in the numerator is the standard length on ⁠${\displaystyle {\mathbb{R}}^m}$⁠. If ${\displaystyle v}$ is a vector starting at ⁠${\displaystyle a}$⁠, then ${\displaystyle f^{\prime }(\mathbf{a})\mathbf{v}}$ is called the pushforward of ${\displaystyle \mathbf{v}}$ by ⁠${\displaystyle f}$⁠.

If the total derivative exists at ⁠${\displaystyle \mathbf{a}}$⁠, then all the partial derivatives and directional derivatives of ${\displaystyle f}$ exist at ⁠${\displaystyle \mathbf{a}}$⁠, and for all ⁠${\displaystyle \mathbf{v}}$⁠, ${\displaystyle f^{\prime }(\mathbf{a})\mathbf{v}}$ is the directional derivative of ${\displaystyle f}$ in the direction ⁠${\displaystyle \mathbf{v}}$⁠. If ${\displaystyle f}$ is written using coordinate functions, so that ⁠${\displaystyle f=(f_1,f_2,\dots ,f_m)}$⁠, then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of ${\displaystyle f}$ at ${\displaystyle \mathbf{a}}$: $${\displaystyle f^{\prime }(\mathbf{a})={\mathrm{Jac}}_{\mathbf{a}}={\left(\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{x}_j}\right)}_{ij}.}$$

Generalizations

Main article: Generalizations of the derivative

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 ${\displaystyle \mathbb{C}}$ to ⁠${\displaystyle \mathbb{C}}$⁠. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. If ${\displaystyle \mathbb{C}}$ is identified with ${\displaystyle {\mathbb{R}}^2}$ by writing a complex number ${\displaystyle z}$ as ⁠${\displaystyle x+iy}$⁠ then a differentiable function from ${\displaystyle \mathbb{C}}$ to ${\displaystyle \mathbb{C}}$ is certainly differentiable as a function from ${\displaystyle {\mathbb{R}}^2}$ to ${\displaystyle {\mathbb{R}}^2}$ (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 ${\displaystyle M}$ is a space that can be approximated near each point ${\displaystyle x}$ by a vector space called its tangent space: the prototypical example is a smooth surface in ⁠${\displaystyle {\mathbb{R}}^3}$⁠. The derivative (or differential) of a (differentiable) map ${\displaystyle f:M\to N}$ between manifolds, at a point ${\displaystyle x}$ in ⁠${\displaystyle M}$⁠, is then a linear map from the tangent space of ${\displaystyle M}$ at ${\displaystyle x}$ to the tangent space of ${\displaystyle N}$ at ⁠${\displaystyle f(x)}$⁠. The derivative function becomes a map between the tangent bundles of ${\displaystyle M}$ and ⁠${\displaystyle N}$⁠. This definition is used in differential geometry.
• Differentiation can also be defined for maps between vector space, such as Banach space, in which those generalizations are the Gateaux derivative and the Fréchet derivative.
• One deficiency of the classical derivative is that very many functions are not 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".
• Properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology; an example is differential algebra. Here, it consists of the derivation of some topics in abstract algebra, such as rings, ideals, field, and so on.
• 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.
• The arithmetic derivative involves the function that is defined for the integers by the prime factorization. This is an analogy with the product rule.