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].
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].
