Answer :
To find the derivative of the function [tex]\( f(x) = (9x^3 - 2)(x + 3) \)[/tex] using the product rule, let's break it down step by step.
1. Identify the Functions:
The product rule states that if you have a function that is the product of two functions, say [tex]\( u(x) \cdot v(x) \)[/tex], then the derivative is given by:
[tex]\[
(uv)' = u'v + uv'
\][/tex]
For our function, let:
[tex]\[
u(x) = 9x^3 - 2 \quad \text{and} \quad v(x) = x + 3
\][/tex]
2. Find the Derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
- The derivative of [tex]\( u(x) = 9x^3 - 2 \)[/tex] is:
[tex]\[
u'(x) = \frac{d}{dx}(9x^3 - 2) = 27x^2
\][/tex]
- The derivative of [tex]\( v(x) = x + 3 \)[/tex] is:
[tex]\[
v'(x) = \frac{d}{dx}(x + 3) = 1
\][/tex]
3. Apply the Product Rule:
Substitute [tex]\( u(x) \)[/tex], [tex]\( u'(x) \)[/tex], [tex]\( v(x) \)[/tex], and [tex]\( v'(x) \)[/tex] into the product rule formula:
[tex]\[
f'(x) = (9x^3 - 2) \cdot 1 + 27x^2 \cdot (x + 3)
\][/tex]
4. Simplify the Expression:
- First term: [tex]\( (9x^3 - 2) \cdot 1 = 9x^3 - 2 \)[/tex]
- Second term: [tex]\( 27x^2(x + 3) = 27x^3 + 81x^2 \)[/tex]
- Combine both terms:
[tex]\[
f'(x) = 9x^3 - 2 + 27x^3 + 81x^2
\][/tex]
- Combine like terms:
[tex]\[
f'(x) = (9x^3 + 27x^3) + 81x^2 - 2 = 36x^3 + 81x^2 - 2
\][/tex]
Therefore, the derivative [tex]\( f'(x) \)[/tex] is:
[tex]\[ f'(x) = 36x^3 + 81x^2 - 2 \][/tex]
And this matches one of the given answer choices.
1. Identify the Functions:
The product rule states that if you have a function that is the product of two functions, say [tex]\( u(x) \cdot v(x) \)[/tex], then the derivative is given by:
[tex]\[
(uv)' = u'v + uv'
\][/tex]
For our function, let:
[tex]\[
u(x) = 9x^3 - 2 \quad \text{and} \quad v(x) = x + 3
\][/tex]
2. Find the Derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
- The derivative of [tex]\( u(x) = 9x^3 - 2 \)[/tex] is:
[tex]\[
u'(x) = \frac{d}{dx}(9x^3 - 2) = 27x^2
\][/tex]
- The derivative of [tex]\( v(x) = x + 3 \)[/tex] is:
[tex]\[
v'(x) = \frac{d}{dx}(x + 3) = 1
\][/tex]
3. Apply the Product Rule:
Substitute [tex]\( u(x) \)[/tex], [tex]\( u'(x) \)[/tex], [tex]\( v(x) \)[/tex], and [tex]\( v'(x) \)[/tex] into the product rule formula:
[tex]\[
f'(x) = (9x^3 - 2) \cdot 1 + 27x^2 \cdot (x + 3)
\][/tex]
4. Simplify the Expression:
- First term: [tex]\( (9x^3 - 2) \cdot 1 = 9x^3 - 2 \)[/tex]
- Second term: [tex]\( 27x^2(x + 3) = 27x^3 + 81x^2 \)[/tex]
- Combine both terms:
[tex]\[
f'(x) = 9x^3 - 2 + 27x^3 + 81x^2
\][/tex]
- Combine like terms:
[tex]\[
f'(x) = (9x^3 + 27x^3) + 81x^2 - 2 = 36x^3 + 81x^2 - 2
\][/tex]
Therefore, the derivative [tex]\( f'(x) \)[/tex] is:
[tex]\[ f'(x) = 36x^3 + 81x^2 - 2 \][/tex]
And this matches one of the given answer choices.
