$\text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. }$
A) $f ^ { tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$
B) $f ^ {tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$
C) $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 4 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$
D) $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan x \right)$
E) $f ^ {tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x \tan ^ { 2 } x \right)$

Question

$\text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. }$ 
A) $f ^ { tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$
B) $f ^ {tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$
C) $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 4 x \tan x + x ^ { 2 } \tan ^ { 2 } x \right)$
D) $f ^ { tt } ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x ^ { 2 } \tan x \right)$
E) $f ^ {tt} ( x ) = x ^ { 2 } \sec x \left( 12 + x ^ { 2 } \sec ^ { 2 } x + 8 x \tan x + x \tan ^ { 2 } x \right)$

Quizplus · Accepted Answer

We use the product rule twice to find the second derivative: 
\begin{align*}
f(x)&=x^4\sec x \
f'(x)&=4x^3\sec x + x^4\sec x 	an x \
f''(x)&=12x^2\sec x + 4x^3\sec x	an x + 4x^3\sec x	an x + x^4\sec x\sec^2 x + x^4\sec x	an^2 x\
&=12x^2\sec x + 8x^3\sec x	an x + x^4\sec x(\sec^2 x + 	an^2 x)\
&=12x^2\sec x + 8x^3\sec x	an x + x^4\sec x(1+	an^2 x)\
&=12x^2\sec x + 8x^3\sec x	an x + x^4\sec x\sec^2 x\
&=x^2\sec x(12 + x^2\sec^2 x + 8x	an x + x^2	an^2 x)
\end{align*} 
Therefore, the correct choice is (B).

Find the second derivative of the function f(x)=x4sec⁡x. \text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. } Find the second derivative of the function f(x)=x4secx.

$\text { Find the second derivative of the function } f ( x ) = x ^ { 4 } \sec x \text {. }$