# Derivative of Constant Multiple/Real/Corollary

## Corollary to Derivative of Constant Multiple

Let $f$ be a real function which is differentiable on $\R$.

Let $c \in \R$ be a constant.

Then:

$\map {\dfrac {\d^n} {\d x^n} } {c \map f x} = c \map {\dfrac {\d^n} {\d x^n} } {\map f x}$

## Proof

By induction: the base case is for $n = 1$ and is proved in Derivative of Constant Multiple.

Now consider $\map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {c \map f x}$, assuming the induction hypothesis $\map {\dfrac {\d^k} {\d x^k} } {c \map f x} = c \map {\dfrac {\d^k} {\d x^k} } {\map f x}$:

 $\displaystyle \map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {c \map f x}$ $=$ $\displaystyle \map {\dfrac \d {\d x} } {\map {\dfrac {\d^k} {\d x^k} } {c \map f x} }$ Definition of Higher Derivative $\displaystyle$ $=$ $\displaystyle \map {\dfrac \d {\d x} } {c \map {\dfrac {\d^k} {\d x^k} } {\map f x} }$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle c \map {\dfrac \d {\d x} } {\map {\dfrac {\d^k} {\d x^k} } {\map f x} }$ Basis for the Induction $\displaystyle$ $=$ $\displaystyle c \map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {\map f x}$ Definition of Higher Derivative

Hence the result by induction.

$\blacksquare$