Given $f ( x ) = e ^ { x }$ , $g ( x ) = 5 ^ { x }$ , and h(x)= g(x)/f(x), find $h ^ { \prime \prime } ( x )$ .
A) $5 ^ { x } e ^ { - x } ( \ln 5 + 1 ) ^ { 2 }$
B) $5 ^ { x } e ^ { - x } ( \ln 5 - 1 ) ^ { 2 }$
C) $5 ^ { x } e ^ { - x } ( \ln 5 ) ^ { 2 }$
D) $- x ( x - 1 ) 5 ^ { x - 2 } e ^ { - x - 2 } ( \ln 5 ) ^ { 2 }$

Question

Quizplus · Accepted Answer

First, we find $h(x) = \frac{g(x)}{f(x)} = \frac{5^x}{e^x} = 5^x e^{-x}$. Then, $h'(x) = 5^x e^{-x}(\ln(5) - 1)$ by applying the product rule and chain rule. Differentiating again to find $h''(x)$, we apply the product rule and chain rule again, resulting in $h''(x) = 5^x e^{-x}(\ln(5) - 1)^2$, which matches option B.

Given f(x)=exf ( x ) = e ^ { x }f(x)=ex

Given $f ( x ) = e ^ { x }$