Question

Let $f\left(x\right)$ be a polynomial in x. The second derivative of $f\left(e^x\right)$ at $x=1$ is

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

Answer 1

The correct option is D $(f^{\prime \prime }\left(e\right)e+f^{\prime }\left(e\right))e$

Let $g\left(x\right)=f\left(e^x\right)$
we know $\left(f\right(g\left(x\right)){)}^{\prime }=f^{\prime }(g\left(x\right))\times g^{\prime }(x)$
First derivative of g(x) $=f^{\prime }\left(e^x\right)e^x$
Second derivative of g(x) $=f^{\prime \prime }\left(e^x\right)e^{2x}+f^{\prime }\left(e^x\right)e^x$
$g^{\prime \prime }\left(1\right)=f^{\prime \prime }\left(e\right)e^2+f^{\prime }\left(e\right)e$