GetWiki

{{Short description|Instantaneous rate of change (mathematics)}}{{other uses|}}{{pp-semi-indef|small=yes}}{{good article}}{{Calculus |differential}}The derivative is a fundamental tool of calculus 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.{{sfn|Stewart|2002|p=129–130}} 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 (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 f(x) is differentiable at a point a of its domain, if its domain contains an open interval containing a , and the limitL=lim_{h to 0}frac{f(a+h)-f(a)}h exists.{{sfnm|1a1=Stewart|1y=2002|1p=127 | 2a1=Strang et al.|2y=2023|2p=220}} This means that, for every positive real number varepsilon, there exists a positive real number delta such that, for every h such that |h| < delta and hne 0 then f(a+h) is defined, and left|L-frac{f(a+h)-f(a)}hright|

• Functions of exponential, natural logarithm, and logarithm with general base:
• : 0
• : frac{d}{dx}ln(x) = frac{1}{x} , for x > 0
• : frac{d}{dx}log_a(x) = frac{1}{xln(a)} , for x, a > 0

• Trigonometric functions:
• : frac{d}{dx}sin(x) = cos(x)
• : frac{d}{dx}cos(x) = -sin(x)
• : frac{d}{dx}tan(x) = sec^2(x) = frac{1}{cos^2(x)} = 1 + tan^2(x)

• Inverse trigonometric functions:
• : frac{d}{dx}arcsin(x) = frac{1}{sqrt{1-x^2}} , for -1 < x < 1
• : frac{d}{dx}arccos(x)= -frac{1}{sqrt{1-x^2}} , for -1 < x < 1
• : frac{d}{dx}arctan(x)= frac{1}{{1+x^2}}

{{anchor|Rules}}Rules for combined functions

Given that the f and g are the functions. The following are some of the most basic rules for deducing the derivative of functions from derivatives of basic functions. For constant rule and sum rule, see {{harvnb|Apostol|1967|p=161, 164}}, respectively. For the product rule, quotient rule, and chain rule, see {{harvnb|Varberg|Purcell|Rigdon|2007|p=111–112, 119}}, respectively. For the special case of the product rule, that is, the product of a constant and a function, see {{harvnb|Varberg|Purcell|Rigdon|2007|p=108–109}}.

• Constant rule: if f is constant, then for all x,
• : f'(x) = 0.
• Sum rule:
• : (alpha f + beta g)' = alpha f' + beta g' for all functions f and g and all real numbers alpha and beta.
• Product rule:
• : (fg)' = f 'g + fg' for all functions f and g. As a special case, this rule includes the fact (alpha f)' = alpha f' whenever alpha is a constant because alpha' f = 0 cdot f = 0 by the constant rule.
• Quotient rule:
• : left(frac{f}{g} right)' = frac{f'g - fg'}{g^2} for all functions f and g at all inputs where {{nowrap|g ≠ 0}}.
• Chain rule for composite functions: If f(x) = h(g(x)), then
• : f'(x) = h'(g(x)) cdot g'(x).

Computation example

The derivative of the function given by f(x) = x^4 + sin left(x^2right) - ln(x) e^x + 7 is end{align} 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 x^2 , x^4 , sin (x) , ln (x) , and exp(x) = e^x , as well as the constant 7 , were also used.

Higher-order derivatives{{anchor|order of derivation|Order}}

Higher order derivatives means that a function is differentiated repeatedly. Given that f is a differentiable function, the derivative of f is the first derivative, denoted as f' . The derivative of f' is the second derivative, denoted as f , and the derivative of f is the third derivative, denoted as f' . By continuing this process, if it exists, the {{nowrap|1= n -}}th derivative as the derivative of the {{nowrap|1= (n - 1) -}}th derivative or the derivative of order n ''. As has been discussed above, the generalization of derivative of a function f may be denoted as f^{(n)} .{hide}sfnm {edih} A function that has k successive derivatives is called k times differentiable. If the {{nowrap|1= k -}}th derivative is continuous, then the function is said to be of differentiability class C^k .{{sfn|Warner|1983|p=5}} A function that has infinitely many derivatives is called infinitely differentiable or smooth.{{sfn|Debnath|Shah|2015|p=40}} 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.{{sfn|Carothers|2000|p=176}}{{anchor|1=Instantaneous rate of change}}In one of its applications, the higher-order derivatives may have specific interpretations 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,{{sfn|Apostol|1967|p=160}} and the third derivative is the jerk.{{sfn|Stewart|2002|p=193}}

In other dimensions

{{See also|Vector calculus|Multivariable calculus}}

Vector-valued functions

A vector-valued function mathbf{y} of a real variable sends real numbers to vectors in some vector space R^n . A vector-valued function can be split up into its coordinate functions y_1(t), y_2(t), dots, y_n(t) , meaning that mathbf{y} = (y_1(t), y_2(t), dots, y_n(t)). This includes, for example, parametric curves in R^2 or R^3 . The coordinate functions are real-valued functions, so the above definition of derivative applies to them. The derivative of mathbf{y}(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,{{sfn|Stewart|2002|p=893}} if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of mathbf{y} exists for every value of t , then mathbf{y} is another vector-valued function.{{sfn|Stewart|2002|p=893}}

Partial derivatives

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 f(x, y, dots) with respect to the variable x is variously denoted by{{block indent | em = 1.2 | text = f_x, f'_x, partial_x f, frac{partial}{partial x}f, or frac{partial f}{partial x},}}among other possibilities.{{sfnm }} It can be thought of as the rate of change of the function in the x-direction.{{sfn|Stewart|2002|p=949}} 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".{{sfnm }} For example, let f(x,y) = x^2 + xy + y^2, then the partial derivative of function f with respect to both variables x and y are, respectively: In general, the partial derivative of a function f(x_1, dots, x_n) in the direction x_i at the point (a_1, dots, a_n) is defined to be:{{sfn|Mathai|Haubold|2017|p=52}}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}.This is fundamental for the study of the functions of several real variables. Let f(x_1, dots, x_n) be such a real-valued function. If all partial derivatives f with respect to x_j are defined at the point (a_1, dots, a_n) , these partial derivatives define the vectornabla 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),which 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 nabla f that maps the point (a_1, dots, a_n) to the vector nabla f(a_1, dots, a_n) . Consequently, the gradient determines a vector field.{{sfn|Gbur|2011|pp=36–37}}

Directional derivatives

If f is a real-valued function on R^n , 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 and y direction. However, they do not 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 mathbf{v} = (v_1,ldots,v_n) , then the directional derivative of f in the direction of mathbf{v} at the point mathbf{x} is:{{sfn|Varberg|Purcell|Rigdon|2007|p=642}} If all the partial derivatives of f exist and are continuous at mathbf{x} , then they determine the directional derivative of f in the direction mathbf{v} by the formula:{{sfn|Guzman|2003|p=35}}

Total derivative, total differential and Jacobian matrix

When f is a function from an open subset of R^n 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. However, 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 mathbf{v} starting at mathbf{a} , the linear approximation formula holds:{{sfn|Davvaz|2023|p=266}}f(mathbf{a} + mathbf{v}) approx f(mathbf{a}) + f'(mathbf{a})mathbf{v}.Similarly with the single-variable derivative, f'(mathbf{a}) is chosen so that the error in this approximation is as small as possible. The total derivative of f at mathbf{a} is the unique linear transformation f'(mathbf{a}) colon R^n to R^m such that{{sfn|Davvaz|2023|p=266}}lim_{mathbf{h}to 0} frac{lVert f(mathbf{a} + mathbf{h}) - (f(mathbf{a}) + f'(mathbf{a})mathbf{h})rVert}{lVertmathbf{h}rVert} = 0.Here mathbf{h} is a vector in R^n , so the norm in the denominator is the standard length on R^n . However, f'(mathbf{a}) mathbf{h} is a vector in R^m , and the norm in the numerator is the standard length on R^m .{{sfn|Davvaz|2023|p=266}} If v is a vector starting at a , then f'(mathbf{a}) mathbf{v} is called the pushforward of mathbf{v} by f .{{sfn|Lee|2013|p=72}}If the total derivative exists at mathbf{a} , then all the partial derivatives and directional derivatives of f exist at mathbf{a} , and for all mathbf{v} , f'(mathbf{a})mathbf{v} is the directional derivative of f in the direction mathbf{v} . If f is written using coordinate functions, so that 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 f at mathbf{a} :{{sfn|Davvaz|2023|p=267}}f'(mathbf{a}) = operatorname{Jac}_{mathbf{a}} = left(frac{partial f_i}{partial x_j}right)_{ij}.

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.{{sfn|Roussos|2014|p=303}} If C is identified with R^2 by writing a complex number z as x+iy, then a differentiable function from C to C is certainly differentiable as a function from R^2 to 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.{{sfn|Gbur|2011|pp=261–264}}
• 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 R^3. The derivative (or differential) of a (differentiable) map f:Mto 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 used in differential geometry.{{sfn|Gray|Abbena|Salamon|2006|p=826}}
• 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.{{harvnb|Azegami|2020}}. See p. 209 for the Gateaux derivative, and p. 211 for 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".{{sfn|Funaro|1992|p=84–85}}
• 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.{{sfn|Kolchin|1973|p=58, 126}}
• 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.{{sfn|Georgiev|2018|p=8}}
• The arithmetic derivative involves the function that is defined for the integers by the prime factorization. This is an analogy with the product rule.{{sfn|Barbeau|1961}}

See also

• Integral

Notes

{{reflist}}

References

• {{Citation

• {{citation

• {{Citation

• JOURNAL

,

• {{citation

}}

• {{Citation

• {{Citation

• {{citation

}}

• {{Citation

}}

• {hide}citation

{edih}

• {{Citation

}}

• {{citation

}}

• {{citation

• {{citation

}}

• {hide}Citation

{edih}

• {hide}Citation

{edih}

• {{citation

}}

• {{citation

}}

• {{citation

}}

• {{citation

}}

• {{citation

}}

• {{citation

}}

• {{citation

}}

• {{citation

}}

• {{Citation

• {{citation

}}

• {{citation

• {{Citation

}}

• {{citation

}}

• {hide}Citation

{edih}

• {{Citation

}}

• {{citation

}}

• {{citation

}}

• {{citation

}}

• {{Citation

}}

• {hide}citation

{edih}

• {{citation

}}

• {{citation

}}

• {{Citation

}}

• {{Citation

}}

• {{Citation

}}

• {{Citation

}}

• {{Citation

}}

External links

{{Sister project links|Differentiation|auto=1|wikt=y|b=y|v=y}}

• {{springer|title=Derivative|id=p/d031260}}
• Khan Academy: "Newton, Leibniz, and Usain Bolt"
• {{MathWorld |title=Derivative |id=Derivative}}
• Online Derivative Calculator from Wolfram Alpha.

{{good article}}{{Calculus topics}}{{Analysis-footer}}{{Authority control}}