Differentiate each of the following from first principle: `xsinx`

Explanation

`f'(x)=lim_(h->0)frac{f(x+h)-f(x)}{h}`

`=lim_(h->0)frac{(x+h)sin(x
+h)-xsinx}{h}`

`=lim_(h->0)frac{x(sin(x+h)-sin x)+hsin (x+h)}{h}`

`=lim_(h->0)frac{x. 2cos (frac{x+h+x}{2})sin(frac{x+h-x}{2})+h sin(x+h)}{h}`

`=lim_(h->0)frac{2x cos(frac{2x+h}{2})sin(frac{h}{2})}{h}+lim_(h->0)frac{h sin(x+h)}{h}`

`=2 x cos(frac{2x+0}{2}).frac{1}{2}+sin(x+0)`

`=x cos x+sin x`