https://file.aiquickdraw.com/mathbot/file/1725932344414ub125al3.jpg

To find the derivative of the function \( f(x) = x^5 \left(3x^4 + 8\right) \), we will use the **product rule** of differentiation. 

The product rule states:
\[
(uv)' = u'v + uv'
\]
where \( u = x^5 \) and \( v = 3x^4 + 8 \).

### Step-by-Step Solution:

1. **Differentiate each part:**

   - \( u = x^5 \), so \( u' = 5x^4 \).
   - \( v = 3x^4 + 8 \), so \( v' = 12x^3 \).

2. **Apply the product rule:**

\[
f'(x) = u'v + uv'
\]

Substitute the values:

\[
f'(x) = (5x^4)(3x^4 + 8) + (x^5)(12x^3)
\]

3. **Simplify the expression:**

\[
f'(x) = 15x^8 + 40x^4 + 12x^8
\]

Combine like terms:

\[
f'(x) = (15x^8 + 12x^8) + 40x^4 = 27x^8 + 40x^4
\]

### Conclusion:

The correct derivative is:

\[
f'(x) = 27x^8 + 40x^4
\]

The expression given in the image, \( 28x^6 - 15x^2 \), appears to be incorrect for this function. The correct answer based on the function provided is \( f'(x) = 27x^8 + 40x^4 \).

Would you like further clarification, or do you have any other questions?

### Related Questions:

1. How do you find the derivative of a product of three functions?
2. What is the chain rule, and when should it be used?
3. How do you find the second derivative of the function \( f(x) = x^5(3x^4 + 8) \)?
4. How would the derivative change if the function were \( f(x) = x^5(3x^4 - 8) \)?
5. What are some common mistakes to avoid when applying the product rule?

#### Tip:
Remember, always simplify expressions step-by-step to avoid mistakes and make calculations easier.

Find the derivative of f(x) = x^5(3x^4 + 8).

Learn how to find the derivative of f(x) = x^5(3x^4 + 8) using the product rule. This problem explores key differentiation techniques involving polynomials. Suitable for Calculus students in Grades 11-12.

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