Evaluate the derivative.
-$$f ^ { \prime } ( - 2 ) \text { if } f ^ { \prime } ( x ) = \left( x ^ { 2 } - 1 \right) ( 4 x ) + \left( b x ^ { 2 } + 1 \right) ( 2 x )$$
A) 96
B) -60
C) 94
D) 68

Question

Quizplus · Accepted Answer

First, simplify the given derivative function: $$f'(x) = (x^2 - 1)(4x) + (bx^2 + 1)(2x).$$ Distribute the terms: $$f'(x) = 4x^3 - 4x + 2bx^3 + 2x.$$ Combine like terms: $$f'(x) = (4 + 2b)x^3 - 2x.$$ To find $f'(-2)$, substitute $x = -2$: $$f'(-2) = (4 + 2b)(-2)^3 - 2(-2).$$ Since the value of $b$ is not given, we assume it's a constant that doesn't affect the calculation for specific $x$ values. Thus, the calculation directly depends on the provided formula and the value of $x$. However, without the specific value of $b$, we can't proceed with a numerical answer. My explanation mistakenly assumed we could calculate without $b$, but since $b$ is part of the equation, we need its value to find the exact numerical answer for $f'(-2)$. Given this oversight, let's correct the approach:To correctly evaluate $f'(-2)$, we need to acknowledge the error in the initial explanation. Without the value of $b$, a specific numerical answer cannot be determined directly from the provided choices. The initial step involves a mistake in the calculation process due to the oversight of the variable $b$. Therefore, the correct approach requires either knowing $b$ or recognizing that the question as posed does not provide enough information for a definitive numerical answer from the choices given.

Evaluate the Derivative f′(−2) if f′(x)=(x2−1)(4x)+(bx2+1)(2x)f ^ { \prime } ( - 2 ) \text { if } f ^ { \prime } ( x ) = \left( x ^ { 2 } - 1 \right) ( 4 x ) + \left( b x ^ { 2 } + 1 \right) ( 2 x )f′(−2) if f′(x)=(x2−1)(4x)+(bx2+1)(2x)

Evaluate the Derivative $f ^ { \prime } ( - 2 ) \text { if } f ^ { \prime } ( x ) = \left( x ^ { 2 } - 1 \right) ( 4 x ) + \left( b x ^ { 2 } + 1 \right) ( 2 x )$