Derivative of Function of Constant Multiple/Corollary

Corollary to 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:

$\map {\dfrac \d {\d x} } {\map f {a x + b} } = a \, \map {\dfrac \d {\map \d {a x + b} } } {\map f {a x + b} }$

Proof

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

\(\text {(1)}: \quad\)

\(\ds \map {\dfrac \d {\d x} } {a x + b}\)

\(=\)

\(\ds \map {\dfrac \d {\d x} } {a x} + \map {\dfrac \d {\d x} } b\)

Sum Rule for Derivatives

\(\ds \)

\(=\)

\(\ds a + 0\)

Derivative of Function of Constant Multiple and Derivative of Constant

\(\ds \)

\(=\)

\(\ds a\)

Next:

\(\ds \map {\dfrac \d {\d x} } {\map f {a x + b} }\)

\(=\)

\(\ds \map {\dfrac \d {\d x} } {a x + b} \, \map {\dfrac \d {\map \d {a x + b} } } {\map f {a x + b} }\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds a \map {\dfrac \d {\map \d {a x + b} } } {\map f {a x + b} }\)

from $(1)$

$\blacksquare$