Let
f(x) = x cos x
∴ \(f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\)
\(=\lim\limits_{h \to 0}\frac{(x+h)cos(x+h)-xcosx}{h}\)
\(=\lim\limits_{h \to 0}\frac{(x+h)[cos\,x\,cosh\,h-sinx.sinh]-xcosx}{h}\)
\(=\lim\limits_{h \to 0}\frac{xcosh.cosx-x\,sinx.sinh+h\,cosx.cosh-h\,sinx.sinh-x\,cosx}{h}\)
\(=\lim\limits_{h \to 0}\frac{xcos\,x.(cos\,h-1)+h[cos\,x.cos\,h-sin\,x.sin\,h-x\,sin\,x.sin\,h}{h}\)
\(=\lim\limits_{h \to 0}\frac{xcos\,x(cos\,h-1)}{h}+\)\(\lim\limits_{h \to 0}(-xsin\,x)(\frac{sin\,h}{h})+\) \(\lim\limits_{h \to 0}cos(x+h)\)
\(=-x\,cos\,x\lim\limits_{h \to 0}\frac{1-cos\,h}{h}-xsinx\lim\limits_{h \to 0}(\frac{sin\,h}{h})\) +\(+\lim\limits_{h \to 0}cos(x+h)\)
\(= -x\,cos\,x.\lim\limits_{h \to 0}\frac{2sin^2\frac{h}{2}}{h\times\frac{h}{4}}\times\frac{h}{4}\)\(-xsinx.1+cosx\)
\(=-x\,cos\,x.\frac{2}{4}.\lim\limits_{h \to 0}\) \((\frac{sin\frac{h}{2}}{\frac{h}{2}})^2\) \(h-x\,sin\,x+cos\,x\)
\(=-x\,cos\,x.\frac{1}{2}.1.0-xsinx+cosx\)
= \(cos\,x - x\,sin\,x\)
