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Encyclopedia > List of differentiation identities

Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus Matrix calculus Mean value theorem For other uses, see Calculus (disambiguation). ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In mathematics, the limit of a function is a fundamental concept in analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices, where it defines the matrix derivative. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates List of differentiation identities For other uses, see Derivative (disambiguation). ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable. ... In calculus, Taylors theorem gives a sequence of approximations of a differentiable function near a given point by polynomials (the Taylor polynomials of that function) whose coefficients depend only on the derivatives of the function at that point. ... In differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change are known. ...
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions This article is about the concept of integrals in calculus. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ... In integral calculus, the use of partial fractions is required to integrate the general rational function. ...

It has been suggested that Techniques for differentiation be merged into this article or section. (Discuss)

The primary operation in differential calculus is finding a derivative. This table lists derivatives of many functions. In the following, f and g are differentiable functions, from the real numbers, and c is a real number. These formulas are sufficient to differentiate any elementary function. Image File history File links Mergefrom. ... This article contains a list of techniques for the differentiation of real functions, categorized by type. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... For other uses, see Derivative (disambiguation). ... This article is about functions in mathematics. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by using...

1 General differentiation rules
2 Derivatives of simple functions
3 Derivatives of exponential and logarithmic functions
4 Derivatives of trigonometric functions
5 Derivatives of hyperbolic functions
6 Derivatives of special functions

General differentiation rules

Main article: Differentiation rules

Linearity: $left({cf}right)' = cf'$; $left({f + g}right)' = f' + g'$
Product rule: $left({fg}right)' = f'g + fg'$
Reciprocal rule: $left(frac{1}{f}right)' = frac{-f'}{f^2}$
Quotient rule: $left({f over g}right)' = {f'g - fg' over g^2}, qquad g ne 0$
Chain rule: $(f circ g)' = (f' circ g)g'$
Derivative of inverse function: $(f^{-1})' =frac{1}{f' circ f^{-1}}$ ,

for any differentiable function f of a real argument and with real values, when the indicated compositions and inverses exist. This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. ... In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the reciprocal rule is a shorthand method of finding the derivative of a function that is the reciprocal of a differentiable function, without using the quotient rule or chain rule. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... A function Æ’ and its inverse Æ’â€“1. ...

Derivatives of simple functions

${d over dx} c = 0$

${d over dx} x = 1$

${d over dx} (cx) = c$

${d over dx} |x| = {x over |x|} = sgn x,qquad x ne 0$

${d over dx} x^c = cx^{c-1} qquad mbox{where both } x^c mbox{ and } cx^{c-1} mbox { are defined}$

${d over dx} left({1 over x}right) = {d over dx} left(x^{-1}right) = -x^{-2} = -{1 over x^2}$

${d over dx} left({1 over x^c}right) = {d over dx} left(x^{-c}right) = -cx^{-c-1} = -{c over x^{c+1}}$

${d over dx} sqrt{x} = {d over dx} x^{1over 2} = {1 over 2} x^{-{1over 2}} = {1 over 2 sqrt{x}}, qquad x > 0$

Derivatives of exponential and logarithmic functions

${d over dx} c^x = {c^x ln c },qquad c > 0$

${d over dx} e^x = e^x$

${d over dx} log_c x = {1 over x ln c},qquad c > 0, c ne 1$

${d over dx} ln x = {1 over x},qquad x > 0$

${d over dx} ln |x| = {1 over x}$

${d over dx} x^x = x^x(1+ln x)$

$(f^g)'=f^g left( g'ln f + frac{g}{f} f' right)$

The exponential function is one of the most important functions in mathematics. ... Look up logarithm in Wiktionary, the free dictionary. ...

Derivatives of trigonometric functions

For more details on this topic, see Differentiation of trigonometric functions.

${d over dx} sin x = cos x$

${d over dx} cos x = -sin x$

${d over dx} tan x = sec^2 x = { 1 over cos^2 x}$

${d over dx} sec x = sec x tan x$

${d over dx} csc x = -csc x cot x$

${d over dx} cot x = -csc^2 x = { -1 over sin^2 x}$

${d over dx} arcsin x = { 1 over sqrt{1 - x^2}}$

${d over dx} arccos x = {-1 over sqrt{1 - x^2}}$

${d over dx} arctan x = { 1 over 1 + x^2}$

${d over dx} arcsec x = { 1 over |x|sqrt{x^2 - 1}}$

${d over dx} arccsc x = {-1 over |x|sqrt{x^2 - 1}}$

${d over dx} arccot x = {-1 over 1 + x^2}$

Sine redirects here. ...

Derivatives of hyperbolic functions

${d over dx} sinh x = cosh x = frac{e^x + e^{-x}}{2}$

${d over dx} cosh x = sinh x = frac{e^x - e^{-x}}{2}$

${d over dx} tanh x = operatorname{sech}^2,x$

${d over dx},operatorname{sech},x = - tanh x,operatorname{sech},x$

${d over dx},operatorname{coth},x = -,operatorname{csch}^2,x$

${d over dx},operatorname{csch},x = -,operatorname{coth},x,operatorname{csch},x$

${d over dx},operatorname{arcsinh},x = { 1 over sqrt{x^2 + 1}}$

${d over dx},operatorname{arccosh},x = { 1 over sqrt{x^2 - 1}}$

${d over dx},operatorname{arctanh},x = { 1 over 1 - x^2}$

${d over dx},operatorname{arcsech},x = { -1 over xsqrt{1 - x^2}}$

${d over dx},operatorname{arccoth},x = { 1 over 1 - x^2}$

${d over dx},operatorname{arccsch},x = {-1 over |x|sqrt{1 + x^2}}$

A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions). ...

Derivatives of special functions

Gamma function In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ...

${d over dx},Gamma(x) = int_0^infty t^{x-1} e^{-t} ln t,dt$

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