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Use first principles and small angle approximations to prove that

\begin{aligned} y & = \sin (x) \\ \Rightarrow \frac{d y}{d x} & = \cos (x) \end{aligned}

By first principles,

\frac{d y}{d x} = lim_{h \to 0} \frac{\sin (x + h) - \sin (x)}{h}

Use the addition formula for $\sin$ :

\sin (A + B) = \sin A \cos B + \cos A \sin B

Because $h \to 0,$ we know that $h$ is small, so $\sin (h)$ and $\cos (h)$ can be replaced by their small angle approximations.