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:

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

Proof

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

\(\ds \map {D_x} {c x}\)

\(=\)

\(\ds c \map {D_x} x + x \map {D_x} c\)

Product Rule for Derivatives

\(\ds \)

\(=\)

\(\ds c + x \map {D_x} c\)

Derivative of Identity Function

\(\ds \)

\(=\)

\(\ds c + 0\)

Derivative of Constant

\(\ds \)

\(=\)

\(\ds c\)

Next:

\(\ds \map {D_x} {\map f {c x} }\)

\(=\)

\(\ds \map {D_x} {c x} \map {D_{c x} } {\map f {c x} }\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds c \map {D_{c x} } {\map f {c x} }\)

from above

$\blacksquare$

Examples

Example: $\sin 2 x$

$\map {\dfrac \d {\d x} } {\sin 2 x} = 2 \cos 2 x$

Example: $\map \cos {a x + b}$

$\map {\dfrac \d {\d x} } {\map \cos {a x + b} } = -a \map \sin {a x + b}$

Example: $\map \sec {a x + b}$

$\map {\dfrac \d {\d x} } {\map \sec {a x + b} } = a \map \sec {a x + b} \map \tan {a x + b}$