Find the derivative of the function $\sin 2x$, by using the first principle w.r to x.

let $f\left(x\right)=\sin 2x$ ;

$f(x+h)=\mathrm{Sin}2(x+h)$

By the first principle

$=\underset{h\to 0}{Lt}2\cdot \cos \left(\frac{4x+2h}{2}\right)\frac{\sin \left(\frac{2h}{2}\right)}{h}$

$=\underset{h\to 0}{Lt}\frac{\sin 2(x+h)-\sin 2x}{h}$

$=\underset{h\to 0}{Lt}\frac{2\cos \frac{(2x+2x+2h)}{2}\sin \left(\frac{2x+2h-2x}{2}\right)}{h}$

$f^l\left(x\right)=2\cdot \underset{h\to 0}{Lt}\cos \left(2x\right)\cdot 1=2\cos 2x$