Derivative of Constant Multiple

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

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

Then:
 * $D_x \left({c f \left({x}\right)}\right) = c D_{x} \left({f \left({x}\right)}\right)$

Corollary

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

Proof of Corollary
By induction: the base case is for $n = 1$ and has been proved as the main result.

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

Hence the result by induction.