Let $f$ be a differentiable function such that $f(1)=2$ and $f^{\prime}(\mathrm{x})=f(\mathrm{x})$ for all $\mathrm{x} \in \mathrm{R}$. If $\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{f}(\mathrm{x}))$, then $h^{\prime}(1)$ is equal to :
Correct Option: , 3
$\frac{\mathrm{f}^{\prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}=1 \forall \mathrm{x} \in \mathrm{R}$
Intergrate \& use $f(1)=2$
$\mathrm{f}(\mathrm{x})=2 \mathrm{e}^{\mathrm{x}-1} \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{e}^{\mathrm{x}-1}$
$h(x)=f(f(x)) \Rightarrow h^{\prime}(x)=f^{\prime}(f(x)) f^{\prime}(x)$
$h^{\prime}(1)=f^{\prime}(f(1)) f^{\prime}(1)$
$=\mathrm{f}^{\prime}(2) \mathrm{f}^{\prime}(1)$
$=2 \mathrm{e} \cdot 2=4 \mathrm{e}$
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