NationMaster - Encyclopedia: Table of derivatives

Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. ... In mathematics, the limit of a function is a fundamental concept in mathematical 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, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... 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 to the average derivative of the section. ...
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives For a non-technical overview of the subject, see Calculus. ... In mathematics, the product rule of calculus, 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, to give a function implicitly is to give an equation that at least in part has the same graph as . ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ...
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution In calculus, the integral of a function is an extension of the concept of a sum. ... 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, possibly 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. ...

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. Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... For a non-technical overview of the subject, see Calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... 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...

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 Derivative of inverse function

General differentiation rules

Main article: Differentiation rules

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

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 mathematics, the product rule of calculus, 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. ...

Derivatives of simple functions

${d over dx} c = 0$

${d over dx} x = 1$

${d over dx} cx = c$

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

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

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

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

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

Derivatives of exponential and logarithmic functions

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

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

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

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

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

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

The exponential function is one of the most important functions in mathematics. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

Derivatives of trigonometric functions

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

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

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

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

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

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

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

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

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

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

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

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

All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...

Derivatives of hyperbolic functions

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

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

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

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

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

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

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

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

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

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

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

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

In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

Derivative of inverse function

${d over dx} (f^{-1}(x))=frac{1}{f'(f^{-1}(x))}$ , for any differentiable function f of a real argument and with real values, when the indicated compositions and inverses exist.

Categories: Differential calculus | Mathematics-related lists | Mathematical tables In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

Results from FactBites:

derivative: Definition, Synonyms and Much More from Answers.com (3377 words)
	Jerk is the derivative (with respect to time) of an object's acceleration, that is, the third derivative (with respect to time) of an object's position, and second derivative (with respect to time) of an object's velocity.
	The common thread is that the derivative at a point serves as a linear approximation of the function at that point.
	Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.

More on Derivatives (2418 words)
	The derivative of a function f at x is geometrically the slope of the tangent line to the graph of f at x.
	If the derivative of f exists at every point x in the domain, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x.
	Points on the graph of a function where the derivative is undefined or equals zero are called critical points or sometimes stationary points (in the case where the derivative equals zero).

More results at FactBites »

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