To determine the instantaneous rate of change of the function [tex]\( f(x) = 6x^2 - x \)[/tex] at the point [tex]\((-1, 7)\)[/tex], we need to find the value of the derivative of [tex]\( f(x) \)[/tex] at [tex]\( x = -1 \)[/tex].
1. Find the derivative of the function:
The function given is [tex]\( f(x) = 6x^2 - x \)[/tex]. To find its derivative, we can use standard differentiation rules.
[tex]\[
f'(x) = \frac{d}{dx}(6x^2) - \frac{d}{dx}x
\][/tex]
Let's differentiate each term separately:
[tex]\[
\frac{d}{dx}(6x^2) = 12x
\][/tex]
[tex]\[
\frac{d}{dx}(-x) = -1
\][/tex]
Thus, the derivative of the function is:
[tex]\[
f'(x) = 12x - 1
\][/tex]
2. Evaluate the derivative at the given point [tex]\( x = -1 \)[/tex]:
We need to find [tex]\( f'(-1) \)[/tex]:
[tex]\[
f'(-1) = 12(-1) - 1
\][/tex]
[tex]\[
f'(-1) = -12 - 1
\][/tex]
[tex]\[
f'(-1) = -13
\][/tex]
Therefore, the instantaneous rate of change of the function [tex]\( f(x) \)[/tex] at the point [tex]\((-1, 7)\)[/tex] is [tex]\(-13\)[/tex].
The correct answer is [tex]\(-13\)[/tex].
