# Derivative of Function of Constant Multiple

## Theorem

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

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

Then:

$\map {D_x} {\map f {c x} } = c \, \map {D_{c x} } {\map f {c x} }$

### Corollary

Let $a, b \in \R$ be constants.

Then:

$D_x \left({f \left({a x + b}\right)}\right) = a D_{a x + b} \left({f \left({a x + b}\right)}\right)$

## Proof

First it is shown that $\map {D_x} {c x} = c$:

 $\displaystyle \map {D_x} {c x}$ $=$ $\displaystyle c \, \map {D_x} x + x \, \map {D_x} c$ Product Rule $\displaystyle$ $=$ $\displaystyle c + x \, \map {D_x} c$ Derivative of Identity Function $\displaystyle$ $=$ $\displaystyle c + 0$ Derivative of Constant $\displaystyle$ $=$ $\displaystyle c$

Next:

 $\displaystyle \map {D_x} {\map f {c x} }$ $=$ $\displaystyle \map {D_x} {c x} \, \map {D_{c x} } {\map f {c x} }$ Chain Rule $\displaystyle$ $=$ $\displaystyle c \, \map {D_{c x} } {\map f {c x} }$ from above

$\blacksquare$