If `f(0)=f(1)=0` , `f^(prime)(1)=f^(prime)(0)=2` and `y=f(e^x)e^(f(x))` , write the value of `(dy)/(dx)` at `x=0` .

Explanation

We have, `f(0)=f(1)=0, f^{prime}(1)=2`

and,


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

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

`Rightarrow frac{d y}{d x}=f(e^{x}) frac{d}{d x} e^{f(x)}+e^{f(x)} frac{d}{d x} f(e^{x})[text { Using product rule ] } `

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

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

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