Derivative of Constant Multiple/Real/Corollary

Corollary to Derivative of Constant Multiple
Let $f$ be a real function which is differentiable on $\R$.

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

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

Proof
By induction: the base case is for $n = 1$ and is proved in Derivative of Constant Multiple.

Now consider $\map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {c \map f x}$, assuming the induction hypothesis $\map {\dfrac {\d^k} {\d x^k} } {c \map f x} = c \map {\dfrac {\d^k} {\d x^k} } {\map f x}$:

Hence the result by induction.