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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
Question
Suppose for a differentiable function $h, h(0)=0$, $\mathrm{h}(1)=1$ and $\mathrm{h}^{\prime}(0)=\mathrm{h}^{\prime}(1)=2$. If $\mathrm{g}(\mathrm{x})=\mathrm{h}\left(\mathrm{e}^{\mathrm{x}}\right) \mathrm{e}^{\mathrm{h}(\mathrm{x})}$, then $g^{\prime}(0)$ is equal to:
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Answer

d
$ g(x)=h\left(e^x\right) \cdot e^{h(x)} $

$ g^{\prime}(x)=h\left(e^x\right) \cdot e^{h(x)} \cdot h^{\prime}(x)+e^{h(x)} h^{\prime}\left(e^x\right) \cdot e^x $

$ g^{\prime}(0)=h(1) e^{h(0)} h^{\prime}(0)+e^{h(0)} h^{\prime}(1) $

$ =2+2=4$

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