Question

Let $f ( x )$ be a polynomial in $x$. Then the second derivative of $f\left(e^{x}\right)$ w.r.t. $x$ is

• A. $f^{\prime \prime}\left(e^{x}\right) \cdot e^{x}+f^{\prime}\left(e^{x}\right)$
• B. $f^{\prime \prime}\left(e^{x}\right) \cdot e^{2 x}+f^{\prime}\left(e^{x}\right) \cdot e^{2 x}$
• C. $f^{\prime \prime}\left(e^{x}\right) e^{2 x}$
• D. $f^{\prime \prime}\left(e^{x}\right) \cdot e^{2 x}+f^{\prime}\left(e^{x}\right) \cdot e^{x}$

Answer 1

Answer: (D)

$\frac{d}{d x}\left(f\left(e^{x}\right)\right)=f^{\prime}\left(e^{x}\right) \cdot e^{x}$
$\frac{d^{2}}{d x^{2}}\left(f\left(e^{x}\right)\right)=e^{x} f^{\prime}\left(e^{x}\right)+e^{2 x} f^{\prime \prime}\left(e^{x}\right)$