Differentiate `x\ tan^(-1)x` from first principles.

Explanation

We have `f^{prime}(x)`


`=lim _{h Rightarrow 0}(frac{f(x+h)-f(x)}{h}) `

`=lim _{h Rightarrow 0}(frac{tan ^{-1}(x+h)-tan ^{-1} x}{h}) `

`=lim _{h Rightarrow 0}(frac{tan ^{-1}(frac{x+h-x}{1+x(x+h)})}{h}) `

`=lim _{h Rightarrow 0}(frac{tan ^{-1}(frac{h}{1+x(x+h)})}{h}) `

`=lim _{h Rightarrow 0}(frac{tan ^{-1}(frac{h+x(x+h)}{h})}{1+x(x+h)}) `

`.=frac{1}{1+x^{2}}`

...