Derivative of the Real Sine Function

Theorem: Derivative of the Real Sine Function

The derivative of the real sine function is the real cosine function.

$$(\sin x{)}^{\prime }=\cos x$$

PROOF

$$\begin{array}{rl}(\sin x{)}^{\prime }& =\underset{\Delta x\to 0}{\lim }\frac{\sin (x+\Delta x)-\sin x}{\Delta x}=\underset{\Delta x\to 0}{\lim }\frac{2\sin \frac{\Delta x}{2}\cos \frac{2x+\Delta x}{2}}{\Delta x}\\ & =\underset{\Delta x\to 0}{\lim }\frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}}\underset{\Delta x\to 0}{\lim }\cos \left(x+\frac{\Delta x}{2}\right)=1\cdot \cos x=\cos x\end{array}$$