Derivative of Cosine Function

Theorem
$$D_x \left({\cos x}\right) = -\sin x$$

Corollary
If $$y = \cos (a x)\,$$, then $$\frac{dy}{dx} = -a \sin (ax)$$.

Proof
From the definition of the cosine function, we have $$\cos x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!}$$.

Then:

$$ $$ $$ $$

The result follows from the definition of the sine function.

Proof of Corollary
Follows directly from Derivative of Function of Constant Multiple.