Factor rule

The factor rule is one of the basic rules of differential calculus in analysis and says that a constant factor is retained when differentiating. It follows directly from the definition of the derivation, but can also be viewed as a special case of the product rule .

rule

If the function is differentiable at this point and is a real number , then the function is also with ${\displaystyle u}$ ${\displaystyle x_{0}}$ ${\displaystyle k}$ ${\displaystyle f}$

${\displaystyle f(x)=k\cdot u(x)}$

differentiable at the point , and it applies ${\displaystyle x_{0}}$

${\displaystyle f'(x_{0})=k\cdot u'(x_{0})\,.}$

example

The function has the derivative function . ${\displaystyle \ u(x)=x^{2}}$ ${\displaystyle \ u'(x)=2x}$

Then it follows from the factor rule that the function has the derivative function . ${\displaystyle \ f(x)=5\ u(x)=5\cdot x^{2}}$ ${\displaystyle f'(x)=5\cdot u'(x)=5\cdot 2x=10x}$

Individual evidence

1. ^ Lothar Papula: Mathematics for engineers and scientists. Part 1. 13th edition. Vieweg + Teubner Verlag, Wiesbaden 2011, ISBN 978-3-8348-1749-5 , p. 331 ( limited preview in the Google book search).