# Derivative of Function of Constant Multiple/Corollary

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## 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 $\map {D_x} {a x + b} = a$:

\(\text {(1)}: \quad\) | \(\displaystyle \map {D_x} {a x + b}\) | \(=\) | \(\displaystyle \map {D_x} {a x} + \map {D_x} b\) | Sum Rule for Derivatives | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a + 0\) | Derivative of Function of Constant Multiple and Derivative of Constant | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a\) |

Next:

\(\displaystyle \map {D_x} {\map f {a x + b} }\) | \(=\) | \(\displaystyle \map {D_x} {a x + b} \, \map {D_{a x + b} } {\map f {a x + b} }\) | Chain Rule for Derivatives | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a \, \map {D_{a x + b} } {\map f {a x + b} }\) | from $(1)$ |

$\blacksquare$