Derivative of Sine Function

Theorem

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

Corollary

 * $D_x \left({\sin \left({a x}\right)}\right) = a \cos \left({a x}\right)$

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

By the ratio test, this series converges for all $x$.

From Derivative of Power Series, we have:


 * $\displaystyle D_x \left({\sin x}\right) = \sum_{n=0}^\infty \left({-1}\right)^n \left({2n+1}\right) \frac {x^{2n}} {\left({2n+1}\right)!}

= \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n}} {\left({2n}\right)!}$

The result follows from the definition of the cosine function.

Proof 3
This proof depends on Derivative of Cosine Function.

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