Differentiate the following functions with respect to `x`: `tan^(-1)(e^x)`

Explanation

Let `y=tan ^{-1}(sqrt{x})`


Differentiating w.r.t. `x`, we get `frac{d y}{d x}=frac{d}{d x}[tan ^{-1}(sqrt{x})]`



`=frac{1}{1+(sqrt{x})^{2}} cdot frac{d}{d x}(sqrt{x}) `


`=frac{1}{1+x} times frac{1}{2 sqrt{x}} `


`=frac{1}{2 sqrt{x}(1+x)}`