NationMaster - Encyclopedia: Reciprocal rule

Topics in calculus
Fundamental theorem \| Function \| Limits of functions \| Continuity \| Mean value theorem \| Vector calculus \| Tensor calculus Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... The fundamental theorem of calculus is of such central importance in calculus that it is called the fundamental theorem for the entire field of study. ... Partial plot of a function f. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... For any function that is continuous on [a, b] and differentiable on (a, b) there exists some c in the interval (a, b) such that the secant joining the endpoints of the interval [a, b] is parallel to the tangent at c. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is a generalized quantity or a certain kind of geometrical entity that includes all the ideas of scalars, vectors, matrices and linear operators. ...
Differentiation
Product rule \| Quotient rule \| Chain rule \| Implicit differentiation \| Taylor's theorem \| Related rates In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ... In mathematics, the product rule of calculus, which is 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 composition of two functions. ... In mathematics, to give an implicit function f is to give the graph of a function, as a relation. ... 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
Integration by substitution \| Integration by parts \| Integration by trigonometric substitution \| Integration by disks \| Integration by cylindrical shells \| Improper integrals \| Lists of integrals In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... 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, trigonometric substitution is the substitution of trigonometric functions for other expressions. ... 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. ... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... 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 hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of arc 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. Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ... Partial plot of a function f. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... 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 composition of two functions. ...

The reciprocal rule states that the derivative of $1 / g (x)$ is given by

$frac{d}{dx}left(frac{1}{g(x)}right) = frac{- g'(x)}{(g(x))^2}$

where $g(x) neq 0.$

1 Proof
- 1.1 From the quotient rule
- 1.2 From the chain rule
2 Examples
3 See also

Proof

From the quotient rule

The reciprocal rule is derived from the quotient rule, with the numerator $f (x) = 1$ . Then, 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. ...

$frac{d}{dx}left(frac{1}{g(x)}right) = frac{d}{dx}left(frac{f(x)}{g(x)}right)$	$= frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$
	$= frac{0cdot g(x) - 1cdot g'(x)}{(g(x))^2}$
	$= frac{- g'(x)}{(g(x))^2}.$

From the chain rule

The reciprocal rule can also be derived from the chain rule. Let $f (x) = x - 1 .$ Then, In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...

$f(g(x))=(g(x))^{-1}=frac{1}{g(x)}$ .

By the chain rule,

$frac{d}{dx}left(frac{1}{g(x)}right) = frac{d}{dx} f(g(x))$	$= f'(g(x))cdot g'(x)$
	$= -(g(x))^{-2} cdot g'(x)$
	$= frac{- g'(x)}{(g(x))^2}.$

Examples

The derivative of $1 / (x 2 + 2 x)$ is:

$frac{d}{dx}left(frac{1}{x^2 + 2x}right) = frac{-2x - 2}{(x^2 + 2x)^2}.$

The derivative of $1 / cos(x)$ (when $cos xnot=0$ ) is:

$frac{d}{dx} left(frac{1}{cos(x) }right) = frac{-sin(x)}{cos^2(x)} = frac{1}{cos(x)} frac{sin(x)}{cos(x)} = sec(x)tan(x).$

For more general examples, see the derivative article. In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ...

Contents

Proof var ACE_AR = {Site: '756743', Size: '300250'};

From the quotient rule

From the chain rule

Examples

See also

Proof