Derivative of Constant Multiple/Real

Theorem

Let $f$ be a real function which is differentiable on $\R$.

Let $c \in \R$ be a constant.

Then:

$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$

Corollary

$\map {\dfrac {\d^n} {\d x^n} } {c \map f x} = c \map {\dfrac {\d^n} {\d x^n} } {\map f x}$

Proof

 $\ds \map {\dfrac \d {\d x} } {c \map f x}$ $=$ $\ds c \map {\dfrac \d {\d x} } {\map f x} + \map f x \map {\dfrac \d {\d x} } c$ Product Rule for Derivatives $\ds$ $=$ $\ds c \map {\dfrac \d {\d x} } {\map f x} + 0$ Derivative of Constant $\ds$ $=$ $\ds c \map {\dfrac \d {\d x} } {\map f x}$

$\blacksquare$