Let,
f(x) = sin2x
∴ \(f'(x) = \lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\)
\( = \lim\limits_{h \to 0}\frac{sin^2(x+h)-sin^2x}{h}\)
\( = \lim\limits_{h \to 0}\frac{\{sin(x+h)+sinx\}\{sin(x+h)-sin\,x\}}{h}\)
\( = \lim\limits_{h \to 0}\frac{(2sin\frac{x+h+x}{2}cos\frac{x+h-x}{2}).(2cos\frac{x+h+x}{2}sin\frac{x+h-x}{2})}{h}\)
\( = \lim\limits_{h \to 0}\frac{4cos\frac{h}{2}sin\frac{h}{2}.cos(x+\frac{h}{2})sin(x+\frac{h}{2})}{h}\)
\( = 2\lim\limits_{h \to 0}\frac {sin \frac{h}{2}}{\frac{h}{2}}\)\( \lim\limits_{h \to 0}\cos\frac{h}{2}cos(x+\frac{h}{2})\)
= 2 ∙ 1 ∙ 1 ∙ cos x ∙ sin x
= 2 cos x sin x
= sin 2x.
