Let,
f(x) = cos(2x2 − 3)
\(∴ f'(x)= \lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\)
\(= \lim\limits_{h \to 0}\frac{cos\{2(x+h)^2-3\}-cos\{2x^2-3\}}{h}\)
\(= \lim\limits_{h \to 0}\frac{-2sin(\frac{2(x+h)^2-3+2x^2-3}{2})\,sin(\frac{2(x+h)^2-3-2x^2+3}{2})}{h}\)
\(=-2 \lim\limits_{h \to 0}\frac{-2sin\frac{2x^2+4xh+2x^2-6}{2}\,sin\frac{2x^2+4xh+2h^2-2x^2}{2}}{h}\)
\(= -2\lim\limits_{h \to 0}\frac{sin(2x^2+h^2+2xh-3)\,sin(h^2+2xh)}{h}\)
\(= -2\lim\limits_{h \to 0}(h+2x)\frac{sin(h^2+2xh)}{h^2+2xh}\,\lim\limits_{h \to 0}sin(2x^2+h^2+2xh-3)\)
\(= -2\lim\limits_{h \to 0}(h+2x)\lim\limits_{h \to 0}\frac{sin\,(h^2+2x+h)}{h^2+2xh}sin(2x^2-3)\)
\(= -2.2x.1.sin(2x^2-3)\)
\(= -4x\,sin(2x^2-3)\)
