Calculate the derivative of the function. $\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }$
A) $g ^ { \prime } ( x ) = \left( - 16 x ^ { 2 } + 8 x + 8 \right) ^ { - 5 }$
B) $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$
C) $g ^ { \prime } ( x ) = - 32 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$
D) $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right)$
E) $g ^ { \prime } ( x ) = - 4 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$

Question

Calculate the derivative of the function.   $\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }$  
A) $g ^ { \prime } ( x ) = \left( - 16 x ^ { 2 } + 8 x + 8 \right) ^ { - 5 }$
B) $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$
C)  $g ^ { \prime } ( x ) = - 32 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$
D)   $g ^ { \prime } ( x ) = - 4 ( 8 x + 2 ) \left( 4 x ^ { 2 } + 2 x + 2 \right)$
E)   $g ^ { \prime } ( x ) = - 4 \left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 5 }$

Quizplus · Accepted Answer

To find the derivative of $(4x^2 + 2x + 2)^{-4}$, we use the chain rule. The outer function is $u^{-4}$ with $u = 4x^2 + 2x + 2$. The derivative of the outer function with respect to $u$ is $-4u^{-5}$, and the derivative of $u$ with respect to $x$ is $8x + 2$. Applying the chain rule, we get $-4(8x + 2)(4x^2 + 2x + 2)^{-5}$.

Calculate the Derivative of the Function (4x2+2x+2)−4\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }(4x2+2x+2)−4

Calculate the Derivative of the Function $\left( 4 x ^ { 2 } + 2 x + 2 \right) ^ { - 4 }$