Derivative of Sine Function

Theorem
If $y = \sin(x)\,$, then $\displaystyle \frac{dy}{dx}=\cos(x)$.

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

Proof
From the definition of the sine function, we have:
 * $\displaystyle \sin x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!}$

From Power Series Differentiable on Interval of Convergence, we have:


 * $\displaystyle D_x \left({\sin x}\right) = \sum_{n=1}^\infty \left({-1}\right)^n \left({2n+1}\right)\frac {x^{2n}}{\left({2n+1}\right)!} = \sum_{n=1}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!}$

The result follows from the definition of the cosine function.

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