In mathematics, the derivative is a fundamental tool that quantifies the sensitivity of 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. 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. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation, two of the most commonly used being Leibniz notation and prime notation. 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. The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances. Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is 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. A function of a real variable displaystylef(x) is differentiable at a point displaystyle a of its domain, if its domain contains an open interval containing displaystyle a, and the limit displaystyleL=limₕₜₒ₀fracf(a+h)-f(a)h exists. This means that, for every positive real number displaystylevarepsilon, there exists a positive real number displaystyledelta such that, for every displaystyle h such that displaystyle|h|<delta and displaystylehneq0 then displaystylef(a+h) is defined, and displaystyleleft|L-fracf(a+h)-f(a)hright|<varepsilon, 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 displaystylefʼ(a), read as "displaystyle f prime of displaystyle a"; or it can be denoted displaystyletextstylefracdfdx(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 displaystylefʼ 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 displaystylefʼ(a) whenever displaystylefʼ(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: displaystylef(x)=x². Then the quotient in the definition of the derivative is displaystylefracf(a+h)-f(a)h=frac(a+h)²-a²h=fraca²+2ah+h²-a²h=2a+h. The division in the last step is valid as long as displaystylehneq0. The closer displaystyle h is to displaystyle0, the closer this expression becomes to the value displaystyle2a. The limit exists, and for every input displaystyle a the limit is displaystyle2a. So, the derivative of the squaring function is the doubling function: displaystylefʼ(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. One way to think of the derivative textstylefracdfdx(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 displaystyle1+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 displaystylef(x) becomes displaystylefʼ(x)=operatornamestleft(fracf(x+dx)-f(x)dxright) for an arbitrary infinitesimal displaystyle dx, where displaystyleoperatornamest denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Taking the squaring function displaystylef(x)=x² as an example again, displaystylebeginalignedfʼ(x)&=operatornamestleft(fracx²+2xcdotdx+(dx)²-x²dxright)&=operatornamestleft(frac2xcdotdx+(dx)²dxright)&=operatornamestleft(frac2xcdot dxdx+frac(dx)²dxright)&=operatornamestleft(2x+dxright)&=2x.endaligned 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 displaystylea+h is on the low part of the step, so the secant line from displaystyle a to displaystylea+h is very steep; as displaystyle h tends to zero, the slope tends to infinity. If displaystyle h is positive, then displaystylea+h is on the high part of the step, so the secant line from displaystyle a to displaystylea+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 displaystylef(x)=|x| is continuous at displaystylex=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 displaystyle0 to displaystyle h is displaystyle-1. This can be seen graphically as a "kink" or a "cusp" in the graph at displaystylex=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 displaystylef(x)=x¹/³ is not differentiable at displaystylex=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, 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. One common symbol for the derivative of a function is Leibniz notation. They are written as the quotient of two differentials displaystyle dy and displaystyle dx, which were introduced by Gottfried Wilhelm Leibniz in 1675. It is still commonly used when the equation displaystyley=f(x) is viewed as a functional relationship between dependent and independent variables. The first derivative is denoted by displaystyletextstylefracdydx, 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, textstylefracdydx=fracddxf(x). Higher derivatives are expressed using the notation textstylefracdⁿydxⁿ for the displaystyle n-th derivative of displaystyley=f(x). These are abbreviations for multiple applications of the derivative operator; for example, textstylefracd²ydx²=fracddxBigl(fracddxf(x)Bigr). 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 displaystyleu=g(x) and displaystyley=f(g(x)) then textstylefracdydx=fracdyducdotfracdudx. Another common notation for differentiation is by using the prime mark in the symbol of a function displaystylef(x). This is known as prime notation, due to Joseph-Louis Lagrange. The first derivative is written as displaystylefʼ(x), read as "displaystyle f prime of displaystyle x, or displaystyleyʼ, read as "displaystyle y prime". Similarly, the second and the third derivatives can be written as displaystylefʼʼ and displaystylefʼʼʼ, 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 displaystylefᵐᵃᵗʰʳᵐⁱᵛ or displaystylef⁽⁴⁾. The latter notation generalizes to yield the notation displaystylef⁽ⁿ⁾ for the displaystyle nth 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 displaystyledoty and displaystyleddoty, 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 displaystyleDf(x) and higher derivatives are written with a superscript, so the displaystyle n-th derivative is displaystyleDⁿf(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 displaystyleu=f(x,y), its partial derivative with respect to displaystyle x can be written displaystyleDₓu or displaystyleDₓf(x,y). Higher partial derivatives can be indicated by superscripts or multiple subscripts, e.g. textstyleDₓyf(x,y)=fracpartialpartial yBigl(fracpartialpartial xf(x,y)Bigr) and displaystyletextstyleDₓ²f(x,y)=fracpartialpartial xBigl(fracpartialpartial xf(x,y)Bigr). 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. 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: displaystylefracddxxᵃ=axᵃ⁻¹ Functions of exponential, natural logarithm, and logarithm with general base: displaystylefracddxeˣ=eˣ displaystylefracddxaˣ=aˣln(a), for displaystylea>0 displaystylefracddxln(x)=frac1x, for displaystylex>0 displaystylefracddxlogₐ(x)=frac1xln(a), for displaystylex,a>0 Trigonometric functions: displaystylefracddxsin(x)=cos(x) displaystylefracddxcos(x)=-sin(x) displaystylefracddxtan(x)=sec²(x)=frac1cos²(x)=1+tan²(x) Inverse trigonometric functions: displaystylefracddxarcsin(x)=frac1sqrt1-x², for displaystyle-1<x<1 displaystylefracddxarccos(x)=-frac1sqrt1-x², for displaystyle-1<x<1 displaystylefracddxarctan(x)=frac11+x² Given that the displaystyle f and displaystyle g are the functions. The following are some of the most basic rules for deducing the derivative of functions from derivatives of basic functions. Constant rule: if displaystyle f is constant, then for all displaystyle x, displaystylefʼ(x)=0. Sum rule: displaystyle(alphaf+beta g)ʼ=alphafʼ+betagʼ for all functions displaystyle f and displaystyle g and all real numbers displaystylealpha and displaystylebeta. Product rule: displaystyle(fg)ʼ=fʼg+fgʼ for all functions displaystyle f and displaystyle g. As a special case, this rule includes the fact displaystyle(alpha f)ʼ=alphafʼ whenever displaystylealpha is a constant because displaystylealphaʼf=0cdotf=0 by the constant rule. Quotient rule: displaystyleleft(fracfgright)ʼ=fracfʼg-fgʼg² for all functions displaystyle f and displaystyle g at all inputs where g ≠ 0. Chain rule for composite functions: If displaystylef(x)=h(g(x)), then displaystylefʼ(x)=hʼ(g(x))cdotgʼ(x). The derivative of the function given by displaystylef(x)=x⁴+sinleft(x²right)-ln(x)eˣ+7 is displaystylebeginalignedfʼ(x)&=4x⁽⁴⁻¹⁾+fracdleft(x²right)dxcosleft(x²right)-fracdleft(lnxright)dxeˣ-ln(x)fracdleft(eˣright)dx+0&=4x³+2xcosleft(x²right)-frac1xeˣ-ln(x)eˣ.endaligned 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 displaystylex², displaystylex⁴, displaystylesin(x), displaystyleln(x), and displaystyleexp(x)=eˣ, as well as the constant displaystyle7, were also used. Higher order derivatives means that a function is differentiated repeatedly. Given that displaystyle f is a differentiable function, the derivative of displaystyle f is the first derivative, denoted as displaystylefʼ. The derivative of displaystylefʼ is the second derivative, denoted as displaystylefʼʼ, and the derivative of displaystylefʼʼ is the third derivative, denoted as displaystylefʼʼʼ. By continuing this process, if it exists, the displaystyle nth derivative as 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 displaystylef⁽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 displaystyleCᵏ. A function that has infinitely many derivatives is called infinitely differentiable or smooth. One example of the infinitely differentiable function is polynomial; differentiate this function repeatedly results the constant function, and the infinitely subsequent derivative of that function are all zero. One application of higher-order derivatives is in physics. Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of the function is the acceleration of an object with respect to time, and the third derivative is the jerk. A vector-valued function displaystylemathbfy of a real variable sends real numbers to vectors in some vector space displaystylemathbbRⁿ. A vector-valued function can be split up into its coordinate functions displaystyley₁(t),y₂(t),dots,yₙ(t), meaning that displaystylemathbfy=(y₁(t),y₂(t),dots,yₙ(t)). This includes, for example, parametric curves in displaystylemathbbR² or displaystylemathbbR³. The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of displaystylemathbfy(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is, displaystylemathbfyʼ(t)=limₕₜₒ₀fracmathbfy(t+h)-mathbfy(t)h, if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of displaystylemathbfy exists for every value of displaystyle t, then displaystylemathbfyʼ is another vector-valued function. 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 displaystylef(x,y,dots) with respect to the variable displaystyle x is variously denoted by 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 displaystylef(x,y)=x²+xy+y², then the partial derivative of function displaystyle f with respect to both variables displaystyle x and displaystyle y are, respectively: displaystylefracpartial fpartial x=2x+y,qquadfracpartial fpartial y=x+2y. In general, the partial derivative of a function displaystylef(x₁,dots,xₙ) in the direction displaystylexᵢ at the point displaystyle(a₁,dots,aₙ) is defined to be: displaystylefracpartial fpartialxᵢ(a₁,ldots,aₙ)=limₕₜₒ₀fracf(a₁,ldots,aᵢ+h,ldots,aₙ)-f(a₁,ldots,aᵢ,ldots,aₙ)h. This is fundamental for the study of the functions of several real variables. Let displaystylef(x₁,dots,xₙ) be such a real-valued function. If all partial derivatives displaystyle f with respect to displaystylexⱼ are defined at the point displaystyle(a₁,dots,aₙ), these partial derivatives define the vector displaystylenablaf(a₁,ldots,aₙ)=left(fracpartial fpartialx₁(a₁,ldots,aₙ),ldots,fracpartial fpartialxₙ(a₁,ldots,aₙ)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 nabla f that maps the point displaystyle(a₁,dots,aₙ) to the vector displaystylenablaf(a₁,dots,aₙ). Consequently, the gradient determines a vector field. If displaystyle f is a real-valued function on displaystylemathbbRⁿ, 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 displaystyley=x. These are measured using directional derivatives. Choose a vector displaystylemathbfv=(v₁,ldots,vₙ), then the directional derivative of displaystyle f in the direction of displaystylemathbfv at the point displaystylemathbfx is: displaystyleDₘₐₜₕbfᵥf(mathbfx)=limₕᵣᵢgₕₜₐᵣᵣₒw₀fracf(mathbfx+hmathbfv)-f(mathbfx)h. If all the partial derivatives of displaystyle f exist and are continuous at displaystylemathbfx, then they determine the directional derivative of displaystyle f in the direction displaystylemathbfv by the formula: displaystyleDₘₐₜₕbfᵥf(mathbfx)=sumⱼ₌₁ⁿvⱼfracpartial fpartialxⱼ. When displaystyle f is a function from an open subset of displaystylemathbbRⁿ to displaystylemathbbRᵐ, 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 displaystylen>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 displaystylemathbfv starting at displaystylemathbfa, the linear approximation formula holds: displaystylef(mathbfa+mathbfv)approxf(mathbfa)+fʼ(mathbfa)mathbfv. Similarly with the single-variable derivative, displaystylefʼ(mathbfa) is chosen so that the error in this approximation is as small as possible. The total derivative of displaystyle f at displaystylemathbfa is the unique linear transformation displaystylefʼ(mathbfa)colonmathbbRⁿtomathbbRᵐ such that displaystylelimₘₐₜₕbfₕₜₒ₀fraclVertf(mathbfa+mathbfh)-(f(mathbfa)+fʼ(mathbfa)mathbfh)rVertlVertmathbfhrVert=0. Here displaystylemathbfh is a vector in displaystylemathbbRⁿ, so the norm in the denominator is the standard length on displaystylemathbbRⁿ. However, displaystylefʼ(mathbfa)mathbfh is a vector in displaystylemathbbRᵐ, and the norm in the numerator is the standard length on displaystylemathbbRᵐ. If displaystyle v is a vector starting at displaystyle a, then displaystylefʼ(mathbfa)mathbfv is called the pushforward of displaystylemathbfv by displaystyle f. If the total derivative exists at displaystylemathbfa, then all the partial derivatives and directional derivatives of displaystyle f exist at displaystylemathbfa, and for all displaystylemathbfv, displaystylefʼ(mathbfa)mathbfv is the directional derivative of displaystyle f in the direction displaystylemathbfv. If displaystyle f is written using coordinate functions, so that displaystylef=(f₁,f₂,dots,fₘ), 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 displaystylemathbfa: displaystylefʼ(mathbfa)=operatornameJacₘₐₜₕbfₐ=left(fracpartialfᵢpartialxⱼright)ᵢⱼ. 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 displaystylemathbbC to displaystylemathbbC. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. If displaystylemathbbC is identified with displaystylemathbbR² by writing a complex number displaystyle z as displaystylex+iy then a differentiable function from displaystylemathbbC to displaystylemathbbC is certainly differentiable as a function from displaystylemathbbR² to displaystylemathbbR² (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 displaystylemathbbR³. The derivative (or differential) of a (differentiable) map displaystylef:Mto 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 displaystylef(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. Covariant derivative Derivation Exterior derivative Functional derivative Integral Lie derivative "Derivative", Encyclopedia of Mathematics, EMS Press, 2001 Khan Academy: "Newton, Leibniz, and Usain Bolt" Weisstein, Eric W. "Derivative". MathWorld. Online Derivative Calculator from Wolfram Alpha.