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 $(1)$

$\blacksquare$