In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.
The quotient rule states that the derivative of h(x) is It is provable in many ways by using other derivative rules.
, then using the quotient rule:
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\left({\frac {e^{x}}{x^{2}}}\right)&={\frac {\left({\frac {d}{dx}}e^{x}\right)(x^{2})-(e^{x})\left({\frac {d}{dx}}x^{2}\right)}{(x^{2})^{2}}}\\&={\frac {(e^{x})(x^{2})-(e^{x})(2x)}{x^{4}}}\\&={\frac {x^{2}e^{x}-2xe^{x}}{x^{4}}}\\&={\frac {xe^{x}-2e^{x}}{x^{3}}}\\&={\frac {e^{x}(x-2)}{x^{3}}}.\end{aligned}}}
The quotient rule can be used to find the derivative of
cos x
cos x
( cos x ) − ( sin x )
( cos x ) ( cos x ) − ( sin x ) ( − sin x )
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\tan x&={\frac {d}{dx}}\left({\frac {\sin x}{\cos x}}\right)\\&={\frac {\left({\frac {d}{dx}}\sin x\right)(\cos x)-(\sin x)\left({\frac {d}{dx}}\cos x\right)}{\cos ^{2}x}}\\&={\frac {(\cos x)(\cos x)-(\sin x)(-\sin x)}{\cos ^{2}x}}\\&={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}\\&={\frac {1}{\cos ^{2}x}}=\sec ^{2}x.\end{aligned}}}
The reciprocal rule is a special case of the quotient rule in which the numerator
Applying the quotient rule gives
Utilizing the chain rule yields the same result.
Applying the definition of the derivative and properties of limits gives the following proof, with the term
added and subtracted to allow splitting and factoring in subsequent steps without affecting the value:
The limit evaluation
is justified by the differentiability of
, implying continuity, which can be expressed as
The product rule then gives
and substituting back for
Then the product rule gives
To evaluate the derivative in the second term, apply the reciprocal rule, or the power rule along with the chain rule:
Substituting the result into the expression gives
Taking the absolute value and natural logarithm of both sides of the equation gives
Applying properties of the absolute value and logarithms,
Taking the logarithmic derivative of both sides,
and substituting back
Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments.
, which justifies taking the absolute value of the functions for logarithmic differentiation.
Implicit differentiation can be used to compute the nth derivative of a quotient (partially in terms of its first n − 1 derivatives).