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Write the correct alternative in the following:

Question:

Write the correct alternative in the following:

Let $f(x)$ be a polynomial. Then, the second order derivative of $f\left(e^{x}\right)$ is

A. $f^{\prime \prime}\left(e^{x}\right) e^{2 x}+f^{\prime}\left(e^{x}\right) e^{x}$

B. $f^{\prime \prime}\left(e^{x}\right) e^{x}+f^{\prime}\left(e^{x}\right)$

C. $f^{\prime \prime}\left(e^{x}\right) e^{2 x}+f^{\prime \prime}\left(e^{x}\right) e^{x}$

D. $f^{\prime \prime}\left(e^{x}\right)$

Solution:

Given:

$\frac{d}{d x}\left[\frac{d}{d x} f\left(e^{x}\right)\right]=?$

Since, $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{f}(\mathrm{g}(\mathrm{x}))=\mathrm{f}^{\prime}(\mathrm{g}(\mathrm{x})) \mathrm{g}^{\prime}(\mathrm{x})$

So, $\frac{d}{d x} f\left(e^{x}\right)=f^{\prime}\left(e^{x}\right) e^{x}$

Also, $\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x})]=\mathrm{f}^{\prime}(\mathrm{x}) \mathrm{g}(\mathrm{x})+\mathrm{g}^{\prime}(\mathrm{x}) \mathrm{f}(\mathrm{x})$

So, $\frac{d}{d x} f^{\prime}\left(e^{x}\right) e^{x}=f^{\prime \prime}(x) e^{x} e^{x}+e^{x} f^{\prime}(x)$

$=f^{\prime \prime}(x) e^{2 x}+e^{x} f^{\prime}(x)$

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