# 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:

$D^n_x \left({c f \left({x}\right)}\right) = c D^n_{x} \left({f \left({x}\right)}\right)$

## Proof

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

Now consider $D^{k + 1}_x \left({c f \left({x}\right)}\right)$, assuming the induction hypothesis $D^k_x \left({c f \left({x}\right)}\right) = c D^k_x \left({f \left({x}\right)}\right)$:

 $\displaystyle D^{k + 1}_x \left({c f \left({x}\right)}\right)$ $=$ $\displaystyle D_x \left({D^k_x \left({c f \left({x}\right)}\right)}\right)$ Definition of Higher Derivative $\displaystyle$ $=$ $\displaystyle D_x \left({c D^k_x \left({f \left({x}\right)}\right)}\right)$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle c D_x \left({D^k_x \left({f \left({x}\right)}\right)}\right)$ Base Case $\displaystyle$ $=$ $\displaystyle c D^{k + 1}_x \left({f \left({x}\right)}\right)$ Definition of Higher Derivative

Hence the result by induction.

$\blacksquare$