Derivative--from Eric Weisstein's World of Mathematics

MathWorld	Calculus and Analysis	Calculus	Differential Calculus

Derivative

The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have. The ``simple'' derivative of a function

with respect to

is denoted either

$\begin{displaymath} {df\over dx}\end{displaymath}$

(1)

(and often written in-line as

). When derivatives are taken with respect to time, they are often denoted using Newton's

overdot notation for fluxions,

$\begin{displaymath} {dx\over dt}=\dot x.\end{displaymath}$

(2)

The derivative of a function with respect to the variable is defined as

$\begin{displaymath} f'(x) \equiv \lim_{h\to 0} {f(x+h)-f(x)\over h}.\end{displaymath}$

(3)

Note that in order for the limit to exist, both $\lim_{h\to 0^+}$ and $\lim_{h\to 0^-}$ must exist and be equal, so the function must be continuous. However, continuity is a necessary but not sufficient condition for differentiability. Since some discontinuous functions can be integrated, in a sense there are ``more'' functions which can be integrated than differentiated. In a letter to Stieltjes, Hermite

wrote, ``I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives.''

A 3-D generalization of the derivative to an arbitrary direction is known as the directional derivative. In general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into ``tangent maps.''

Simple derivatives of some simple functions follow.

${d\over dx} x^n = nx^{n-1}$ (4)

${d\over dx} \ln\vert x\vert = {1\over x}$ (5)

${d\over dx} \sin x = \cos x$ (6)

${d\over dx} \cos x = -\sin x$ (7)

${d\over dx} \tan x = {d\over dx}\left({\sin x\over \cos x}\right)={\cos x\cos x-\sin x(-\sin x)\over \cos ^2 x} = {1\over \cos ^2x} = \sec^2 x$ (8)

${d\over dx} \csc x = {d\over dx}(\sin x)^{-1} = -(\sin x)^{-2} \cos x =-{\cos x\over \sin ^2 x} = -\csc x\cot x$ (9)

${d\over dx} \sec x = {d\over dx}(\cos x)^{-1} = -(\cos x)^{-2}(-\sin x) = {\sin x\over \cos ^2 x} = \sec x\tan x$ (10)

${d\over dx} \cot x = {d\over dx}\left({\cos x\over \sin x}\right)= {\sin x(-\sin x)-\cos x\cos x\over\sin^2x} = - {1\over\sin^2 x} = -\csc^2 x$ (11)

${d\over dx} e^x = e^x$ (12)

${d\over dx} a^x = {d\over dx}e^{\ln a^x} = {d\over dx}e^{x\ln a} = (\ln a) e^{x\ln a} = (\ln a) a^x$ (13)

${d\over dx} \sin^{-1}x = {1\over \sqrt{1-x^2}}$ (14)

${d\over dx} \cos^{-1}x = - {1\over \sqrt{1-x^2}}$ (15)

${d\over dx} \tan^{-1}x = {1\over 1+x^2}$ (16)

${d\over dx} \cot^{-1}x = - {1\over 1+x^2}$ (17)

${d\over dx} \sec^{-1}x = {1\over x\sqrt{x^2-1}}$ (18)

${d\over dx} \csc^{-1}x = - {1\over x\sqrt{x^2-1}}$ (19)

${d\over dx} \sinh x = \cosh x$ (20)

${d\over dx} \cosh x = \sinh x$ (21)

${d\over dx} \tanh x = \mathop{\rm sech}\nolimits ^2 x$ (22)

${d\over dx} \coth x = -\mathop{\rm csch}\nolimits ^2 x$ (23)

${d\over dx} \mathop{\rm sech}\nolimits x = -\mathop{\rm sech}\nolimits x\tanh x$ (24)

${d\over dx} \mathop{\rm csch}\nolimits x = -\mathop{\rm csch}\nolimits x\coth x$ (25)

${d\over dx} \mathop{\rm sn}\nolimits x=\mathop{\rm cn}\nolimits x\mathop{\rm dn}\nolimits x$ (26)

${d\over dx} \mathop{\rm cn}\nolimits x=-\mathop{\rm sn}\nolimits x\mathop{\rm dn}\nolimits x$ (27)

${d\over dx} \mathop{\rm dn}\nolimits x=-k^2\mathop{\rm sn}\nolimits x\mathop{\rm cn}\nolimits x.$ (28)

where $\mathop{\rm sn}\nolimits (x)\equiv \mathop{\rm sn}\nolimits (x,k)$ , $\mathop{\rm cn}\nolimits (x)\equiv \mathop{\rm cn}\nolimits (x,k)$ , etc. are Jacobi elliptic functions, and the product rule and quotient rule have been used extensively to expand the derivatives.

Derivatives of sums are equal to the sum of derivatives so that

$\begin{displaymath}[f(x)+\ldots+h(x)]'=f'(x)+\ldots+h'(x). \end{displaymath}$

(29)

In addition, if

is a constant,

$\begin{displaymath} {d\over dx}[c f(x)]=cf'(x).\end{displaymath}$

(30)

Furthermore, the product rule for differentiation states

$\begin{displaymath} {d\over dx}[f(x)g(x)] = f(x)g'(x)+f'(x)g(x),\end{displaymath}$

(31)

where

denotes the derivative of

with respect to

. This derivative rule can be applied iteratively to yield derivate rules for products of three or more functions, for example,

$\displaystyle [fgh]'$	$\textstyle =$	$\displaystyle (fg)h'+(fg)'h=fgh'+(fg'+f'g)h$
	$\textstyle =$	$\displaystyle f'gh+fg'h+fgh'.$	(32)

The quotient rule

$\begin{displaymath} {d\over dx} \left[{f(x)\over g(x)}\right]= {g(x)f'(x)-f(x)g'(x)\over [g(x)]^2}\end{displaymath}$

(33)

is another important rule which is very useful in computing derivatives. Other important rules for computing derivatives include the chain rule and power rule.

Miscellaneous other derivative identities include

$\begin{displaymath} {dy\over dx} = {{dy\over dt}\over {dx\over dt}}\end{displaymath}$

(34)

$\begin{displaymath} {dy\over dx} = {1\over {dx\over dy}}.\end{displaymath}$

(35)

, where

is a constant, then

$\begin{displaymath} dF = {\partial F\over\partial y} dy + {\partial F\over\partial x} dx = 0,\end{displaymath}$

(36)

$\begin{displaymath} {dy\over dx} = - {{\partial F\over\partial x}\over{\partial F\over\partial y}}.\end{displaymath}$

(37)

A vector derivative of a vector function

$\begin{displaymath} \mathbf{X}(t) \equiv\left[{\matrix{x_1(t)\cr x_2(t)\cr \vdots\cr x_k(t)\cr}}\right]\end{displaymath}$

(38)

can be defined by

$\begin{displaymath} {d\mathbf{X}\over dt} = \left[{\matrix{{dx_1\over dt}\cr {dx_2\over dt}\cr \vdots\cr {dx_k\over dt}\cr}}\right].\end{displaymath}$

(39)

Blancmange Function, Carathéodory Derivative, Chain Rule, Comma Derivative, Convective Derivative, Covariant Derivative, Directional Derivative, Euler-Lagrange Derivative, Fluxion, Fractional Calculus, Fréchet Derivative, Lagrangian Derivative, Lie Derivative, Logarithmic Derivative, Pincherle Derivative, Power Rule, Product Rule, q-Series, Quotient Rule, Schwarzian Derivative, Semicolon Derivative, Weierstrass Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999.

Beyer, W. H. ``Derivatives.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 229-232, 1987.

Griewank, A. Principles and Techniques of Algorithmic Differentiation. Philadelphia, PA: SIAM, 2000.