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Answer :
We start with the function
[tex]$$
f(x) = \left(2x^3 + 2\right)^5.
$$[/tex]
Step 1. Introduce a substitution
Let
[tex]$$
u = 2x^3 + 2.
$$[/tex]
Then
[tex]$$
f(x) = u^5.
$$[/tex]
Step 2. Differentiate using the Chain Rule
The derivative of [tex]$f(x)$[/tex] with respect to [tex]$x$[/tex] is given by
[tex]$$
f'(x) = \frac{d}{dx}\left(u^5\right) = 5u^4 \cdot \frac{du}{dx}.
$$[/tex]
Step 3. Differentiate [tex]$u$[/tex] with respect to [tex]$x$[/tex]
Since
[tex]$$
u = 2x^3 + 2,
$$[/tex]
its derivative is
[tex]$$
\frac{du}{dx} = 6x^2.
$$[/tex]
Step 4. Substitute back into the derivative
Thus,
[tex]$$
f'(x) = 5u^4 \cdot 6x^2 = 30x^2 \left(2x^3 + 2\right)^4.
$$[/tex]
Step 5. Expressing in an Alternative Form
Notice that [tex]$2x^3 + 2$[/tex] can be factored as
[tex]$$
2x^3 + 2 = 2\left(x^3+1\right).
$$[/tex]
Then,
[tex]$$
\left(2x^3+2\right)^4 = \left[2\left(x^3+1\right)\right]^4 = 16\left(x^3+1\right)^4.
$$[/tex]
Substituting back, we have
[tex]$$
f'(x) = 30x^2 \cdot 16\left(x^3+1\right)^4 = 480x^2 \left(x^3+1\right)^4.
$$[/tex]
Step 6. Matching with the Given Options
The expression we obtained for the derivative is
[tex]$$
480x^2 \left(x^3+1\right)^4.
$$[/tex]
None of the provided options match this result since the forms given are:
- (a) [tex]$25 x^2\left(2x^3+2\right)^5$[/tex]
- (b) [tex]$20 x^2\left(2x^3+2\right)^4$[/tex]
- (c) [tex]$20 x^2\left(2x^3+2\right)^5$[/tex]
- (e) [tex]$25 x^2\left(2x^3+2\right)^4$[/tex]
In our result, the constant factor, when comparing with a form like [tex]$C\, x^2\left(2x^3+2\right)^4$[/tex], is [tex]$30$[/tex], not matching [tex]$20$[/tex] or [tex]$25$[/tex]. Hence, none of the options is equal to
[tex]$$
30x^2 \left(2x^3+2\right)^4.
$$[/tex]
Final Conclusion
Since the computed derivative
[tex]$$
f'(x) = 480x^2 \left(x^3+1\right)^4 \quad \text{or equivalently} \quad 30x^2\left(2x^3+2\right)^4
$$[/tex]
does not match any of the given options, the correct choice is:
[tex]$$
\textbf{No Correct Answer.}
$$[/tex]
[tex]$$
f(x) = \left(2x^3 + 2\right)^5.
$$[/tex]
Step 1. Introduce a substitution
Let
[tex]$$
u = 2x^3 + 2.
$$[/tex]
Then
[tex]$$
f(x) = u^5.
$$[/tex]
Step 2. Differentiate using the Chain Rule
The derivative of [tex]$f(x)$[/tex] with respect to [tex]$x$[/tex] is given by
[tex]$$
f'(x) = \frac{d}{dx}\left(u^5\right) = 5u^4 \cdot \frac{du}{dx}.
$$[/tex]
Step 3. Differentiate [tex]$u$[/tex] with respect to [tex]$x$[/tex]
Since
[tex]$$
u = 2x^3 + 2,
$$[/tex]
its derivative is
[tex]$$
\frac{du}{dx} = 6x^2.
$$[/tex]
Step 4. Substitute back into the derivative
Thus,
[tex]$$
f'(x) = 5u^4 \cdot 6x^2 = 30x^2 \left(2x^3 + 2\right)^4.
$$[/tex]
Step 5. Expressing in an Alternative Form
Notice that [tex]$2x^3 + 2$[/tex] can be factored as
[tex]$$
2x^3 + 2 = 2\left(x^3+1\right).
$$[/tex]
Then,
[tex]$$
\left(2x^3+2\right)^4 = \left[2\left(x^3+1\right)\right]^4 = 16\left(x^3+1\right)^4.
$$[/tex]
Substituting back, we have
[tex]$$
f'(x) = 30x^2 \cdot 16\left(x^3+1\right)^4 = 480x^2 \left(x^3+1\right)^4.
$$[/tex]
Step 6. Matching with the Given Options
The expression we obtained for the derivative is
[tex]$$
480x^2 \left(x^3+1\right)^4.
$$[/tex]
None of the provided options match this result since the forms given are:
- (a) [tex]$25 x^2\left(2x^3+2\right)^5$[/tex]
- (b) [tex]$20 x^2\left(2x^3+2\right)^4$[/tex]
- (c) [tex]$20 x^2\left(2x^3+2\right)^5$[/tex]
- (e) [tex]$25 x^2\left(2x^3+2\right)^4$[/tex]
In our result, the constant factor, when comparing with a form like [tex]$C\, x^2\left(2x^3+2\right)^4$[/tex], is [tex]$30$[/tex], not matching [tex]$20$[/tex] or [tex]$25$[/tex]. Hence, none of the options is equal to
[tex]$$
30x^2 \left(2x^3+2\right)^4.
$$[/tex]
Final Conclusion
Since the computed derivative
[tex]$$
f'(x) = 480x^2 \left(x^3+1\right)^4 \quad \text{or equivalently} \quad 30x^2\left(2x^3+2\right)^4
$$[/tex]
does not match any of the given options, the correct choice is:
[tex]$$
\textbf{No Correct Answer.}
$$[/tex]
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