# Derivative of Function of Constant Multiple/Corollary

## Corollary of Derivative of Function of Constant Multiple

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

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

Then:

$\map {D_x} {\map f {a x + b} } = a \, \map {D_{a x + b} } {\map f {a x + b} }$

## Proof

First it is shown that $\map {D_x} {a x + b} = a$:

 $\text {(1)}: \quad$ $\ds \map {D_x} {a x + b}$ $=$ $\ds \map {D_x} {a x} + \map {D_x} b$ Sum Rule for Derivatives $\ds$ $=$ $\ds a + 0$ Derivative of Function of Constant Multiple and Derivative of Constant $\ds$ $=$ $\ds a$

Next:

 $\ds \map {D_x} {\map f {a x + b} }$ $=$ $\ds \map {D_x} {a x + b} \, \map {D_{a x + b} } {\map f {a x + b} }$ Chain Rule for Derivatives $\ds$ $=$ $\ds a \, \map {D_{a x + b} } {\map f {a x + b} }$ from $(1)$

$\blacksquare$