Derivative of Sine Function/Proof 1

Theorem

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

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

From Power Series over Factorial, this series converges for all $x$.

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

The result follows from the definition of the cosine function.