Quotient rule

The quotient rule is a fundamental rule of differential calculus . It traces the calculation of the derivative of a quotient of functions back to the calculation of the derivative of the individual functions.

If the functions and are differentiable from an interval D into the real or complex numbers at the point with , then the function f is also with ${\displaystyle u(x)}$ ${\displaystyle v(x)}$ ${\displaystyle x=x_{a}}$ ${\displaystyle v(x_{a})\neq 0}$

${\displaystyle f(x)={\frac {u(x)}{v(x)}}}$

differentiable at the point and the following applies: ${\displaystyle x_{a}}$

${\displaystyle f'(x_{a})={\frac {u'(x_{a})\cdot v(x_{a})-u(x_{a})\cdot v'(x_{a})}{(v(x_{a}))^{2}}}}$.

In short:

${\displaystyle \left({\frac {u}{v}}\right)'={\frac {u'v-uv'}{v^{2}}}}$

Derivation

The quotient can be interpreted as the slope in a slope triangle whose legs are u (x) and v (x) (see figure). If x increases by Δx, u changes by Δu and v by Δv. The change in slope is then ${\displaystyle u(x) \over v(x)}$

${\displaystyle {\begin{aligned}{\Delta \left({u \over v}\right)}&={{{u+\Delta u} \over {v+\Delta v}}-{u \over v}}\\&={{(u+\Delta u)\cdot v-u\cdot (v+\Delta v)} \over {(v+\Delta v)\cdot v}}\end{aligned}}}$

${\displaystyle ={{\Delta u\cdot v-u\cdot \Delta v} \over {v^{2}+\Delta v\cdot v}}}$

If one divides by Δx, it follows

${\displaystyle {{\Delta \left({u \over v}\right)} \over {\Delta x}}={{{\Delta u \over \Delta x}\cdot v-u\cdot {\Delta v \over \Delta x}} \over {v^{2}+\Delta v\cdot v}}}$

If one now forms Limes Δx against 0, then becomes

${\displaystyle {\left({u \over v}\right)'}={{u'\cdot v-u\cdot v'} \over {v^{2}}}}$

as claimed.

example

If you use the short notation , you get, for example, the derivation of the following function: ${\displaystyle \left({\frac {u}{\color {Blue}v}}\right)'={\frac {u'\color {Blue}v\color {Black}-u\color {Blue}v'}{\color {Blue}v\color {Black}^{2}}}}$

${\displaystyle {\begin{aligned}f(x)&={\frac {x^{2}-1}{\color {Blue}2-3x}}\\f'(x)&={\frac {2x\cdot \color {Blue}(2-3x)\color {Black}-(x^{2}-1)\cdot \color {Blue}(-3)}{(\color {Blue}2-3x\color {Black})^{2}}}\\\end{aligned}}}$

When multiplied it results ${\displaystyle f'(x)={\frac {-3x^{2}+4x-3}{(2-3x)^{2}}}}$

Further derivations

Given After the product rule applies: ${\displaystyle f(x)={\frac {u(x)}{v(x)}}.}$

${\displaystyle {\begin{aligned}f'(x)&=\left(u(x)\cdot {\frac {1}{v(x)}}\right)'\\&=u'(x){\frac {1}{v(x)}}+u(x)\left({\frac {1}{v(x)}}\right)'.\end{aligned}}}$

According to the reciprocal value rule (results e.g. directly or with the help of the chain rule )

${\displaystyle \left({\frac {1}{v(x)}}\right)'=-{\frac {v'(x)}{v^{2}(x)}}.}$

follows:

${\displaystyle {\begin{aligned}f'(x)&=u'(x){\frac {1}{v(x)}}+u(x)\left(-{\frac {v'(x)}{v^{2}(x)}}\right)\\&={\frac {u'(x)v(x)-u(x)v'(x)}{v^{2}(x)}}.\end{aligned}}}$

An alternative derivation is only possible with the product rule by deriving the function equation . However, it is implicitly assumed here that it has a derivative at all, that is, that it exists. ${\displaystyle f(x)\cdot v(x)=u(x)}$ ${\displaystyle f(x)}$ ${\displaystyle f'(x)}$

${\displaystyle f'(x)\cdot v(x)+f(x)\cdot v'(x)=u'(x)}$

consequently:

${\displaystyle {\begin{aligned}f'(x)&={\frac {u'(x)}{v(x)}}-{\frac {u(x)}{v(x)}}\cdot {\frac {v'(x)}{v(x)}}\\&={\frac {u'(x)v(x)-u(x)v'(x)}{v^{2}(x)}}.\end{aligned}}}$

literature

The quotient rule for functions is explained in almost every book that deals with differential calculus in a general way. Some specific examples are:

• Otto Forster : Analysis 1. Differential and integral calculus of a variable. 7th edition. Vieweg, Braunschweig 2004, ISBN 3-528-67224-2 , pp. 155–157 ( excerpt (Google) )
• Konrad Königsberger : Analysis 1 . Springer, Berlin 2004, ISBN 3-540-41282-4 , p. 129
• Harro Heuser : Textbook of Analysis. Part 1 . Vieweg + Teubner, Wiesbaden 1980, ISBN 3-519-02221-4 (17th updated edition. Ibid. 2009, ISBN 978-3-8348-0777-9 ), pp. 270-271 ( excerpt (Google) )

Web links

• Quotient rule on Wikibooks