Reciprocal rule

The reciprocal rule or reciprocal value rule is used to derive mathematical functions of the form

${\displaystyle f(x)={\frac {1}{v(x)}}\,.}$

If the function from an interval into the real or complex numbers is differentiable at the point with , then the function at the point is also differentiable and according to the chain rule, the following applies for the derivative: ${\displaystyle v(x)}$ ${\displaystyle D}$ ${\displaystyle x_{a}}$ ${\displaystyle v(x_{a})\neq 0}$ ${\displaystyle f}$ ${\displaystyle x_{a}}$

${\displaystyle f'(x_{a})=\left[{\frac {1}{v}}\right]'(x_{a})=\left[v^{-1}\right]'(x_{a})=(-1)\cdot v^{-2}(x_{a})\cdot v'(x_{a})=-{\frac {v'(x_{a})}{(v(x_{a}))^{2}}}\,}$

The reciprocal rule is as follows in short form:

${\displaystyle \left[{\frac {1}{v}}\right]'=-{\frac {v'}{v^{2}}}\,.}$

The reciprocal rule can also be a special case of the quotient rule to be interpreted. ${\displaystyle u(x)=1}$

example

The derivative of the function

${\displaystyle f(x)={\frac {1}{\sin(x)}}}$

is calculated at all points where is according to the above reciprocal rule ${\displaystyle \sin(x)\neq 0}$

${\displaystyle f'(x)=-{\frac {\cos(x)}{\sin ^{2}(x)}}}$,

because the cosine function is the derivative of the sine function .

Individual evidence

1. ↑ Harro Heuser: Textbook of Analysis. Part 1. 17th edition. Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-0777-9 , p. 271.
2. ↑ Reciprocal value rule for derivatives. In: Formelsammlung-Mathe.de. Retrieved August 15, 2019 .