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
$$ $$ $$

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

By definition of higher derivatives, we have: $$D^{k+1}_x \left({c f \left({x}\right)}\right) = D_x \left({D^k_x \left({c f \left({x}\right)}\right)}\right)$$

$$ $$ $$ $$

Hence the result by induction.