If `f(1)=4` , `f^(prime)(1)=2` , find the value of the derivative of `log(f(e^x))` with respect to `x` at the point `x=0` .

Explanation

We have, `f(1)=4 and f^{prime}(1)=2`

Let `y=log {f(e^{x})}`


`Rightarrow frac{d y}{d x}=frac{d}{d x}[log {f(e^{x})}] `

`Rightarrow frac{d y}{d x}=frac{1}{f(e^{x})} times frac{d}{d x}{f(e^{x})} `

`Rightarrow frac{d y}{d x}=frac{1}{f(e^{x})} times f^{prime}(e^{x}) times frac{d}{d x}(e^{x})`

`Rightarrow frac{d y}{d x}=frac{e^{x} f^{prime}(e^{x})}{f(e^{x})} `

`text { Putting } x=0, text { we get } `

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