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Let [tex]f[/tex] be the function given by [tex]f(x) = 5x^3 - 3x - 7[/tex]. What is the value of [tex]f^{\prime}(-2)[/tex]?

A. -114
B. 17
C. 50
D. 57

Answer :

To find the value of [tex]\( f'(-2) \)[/tex] for the function [tex]\( f(x) = 5x^3 - 3x - 7 \)[/tex], we need to follow these steps:

1. Find the derivative [tex]\( f'(x) \)[/tex]:

The derivative of a function gives us the rate at which the function's value changes. For a polynomial function like [tex]\( f(x) = 5x^3 - 3x - 7 \)[/tex], we can differentiate each term individually.

[tex]\[
\text{Derivative of } 5x^3 = 15x^2
\][/tex]

[tex]\[
\text{Derivative of } -3x = -3
\][/tex]

[tex]\[
\text{Derivative of a constant } -7 = 0
\][/tex]

So, the derivative [tex]\( f'(x) \)[/tex] is:

[tex]\[
f'(x) = 15x^2 - 3
\][/tex]

2. Evaluate [tex]\( f'(-2) \)[/tex]:

Substitute [tex]\( x = -2 \)[/tex] into the derivative:

[tex]\[
f'(-2) = 15(-2)^2 - 3
\][/tex]

First, calculate [tex]\((-2)^2\)[/tex]:

[tex]\[
(-2)^2 = 4
\][/tex]

Then, multiply by 15:

[tex]\[
15 \times 4 = 60
\][/tex]

Finally, subtract 3:

[tex]\[
60 - 3 = 57
\][/tex]

The value of [tex]\( f'(-2) \)[/tex] is [tex]\(\boxed{57}\)[/tex].

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