Derivative of Function of Constant Multiple
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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: $\map \sec {a x + b}$
- $\map {\dfrac \d {\d x} } {\map \sec {a x + b} } = a \map \sec {a x + b} \map \tan {a x + b}$