Find the second derivative of the function $f ( x ) = 7 ^ { x }$ .
A) $f ^ { \prime \prime } ( x ) = ( \ln 7 ) 7 ^ { x }$
B) $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 3 } 7 ^ { x }$
C) $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 2 } 7 ^ { x }$
D) $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 2 } 7 ^ { 2 x }$
E) $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 3 } 7 ^ { 3 x }$

Question

Quizplus · Accepted Answer

The first derivative of $f(x) = 7^x$ is $f'(x) = (\ln 7)7^x$, using the rule for the derivative of an exponential function $a^x$, which is $f'(x) = \ln(a) \cdot a^x$. Taking the derivative again to find the second derivative, we apply the same rule, resulting in $f''(x) = (\ln 7)^2 7^x$, because the derivative of $(\ln 7)7^x$ involves another multiplication by $\ln 7$.

Find the Second Derivative of the Function f(x)=7xf ( x ) = 7 ^ { x }f(x)=7x

Find the Second Derivative of the Function $f ( x ) = 7 ^ { x }$