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To find \( f'(x) \) for the function given, let's first identify the function and then take the derivative.

The function \( h(x) = x^3 f(x) \) is a product of two functions: \( x^3 \) and \( f(x) \). We can use the **product rule** for differentiation, which is stated as:

\[
\frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]

Here, \( u(x) = x^3 \) and \( v(x) = f(x) \). Now let's compute the derivative:

1. Derivative of \( u(x) = x^3 \) is:
   \[
   u'(x) = 3x^2
   \]

2. The derivative of \( v(x) = f(x) \) is \( v'(x) = f'(x) \).

Now, applying the product rule:

\[
h'(x) = 3x^2 f(x) + x^3 f'(x)
\]

Thus, the derivative of \( h(x) \) is:

\[
h'(x) = 3x^2 f(x) + x^3 f'(x)
\]

If you're looking specifically for \( f'(x) \), this equation relates \( h'(x) \) to \( f'(x) \). Do you have any further information, such as a value for \( h'(x) \), or should we leave the expression in this form?

Let me know if you'd like further clarification! Here are some related questions:

1. How does the product rule apply to more than two functions?
2. How would we differentiate if we had a composition of functions, like \( h(x) = f(g(x)) \)?
3. What happens if we differentiate \( f(x) \) twice? Can we apply the product rule twice in succession?
4. How would the chain rule be applied if the function had nested terms, e.g., \( h(x) = (x^3 f(x))^2 \)?
5. Can we use implicit differentiation to solve this equation if \( f(x) \) were implicitly defined?

**Tip:** Always ensure to simplify after applying the product or chain rule, as the final expression can often be reduced to a more usable form.

Learn how to find the derivative of the function h(x) = x^3 f(x) using the product rule of differentiation. This problem involves key calculus concepts such as the product rule, derivatives of polynomials, and functions. Suitable for high school calculus students in Grades 11-12.

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