Derivative of Cosine Function

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

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

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

Then:

The result follows from the definition of the sine function.

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