https://file.aiquickdraw.com/mathbot/file/1726800131241dknxnab1.jpg

The equation given in the image is:

\[
h(x) = x^3 f(x)
\]

To find \( h'(x) \), the derivative of \( h(x) \), we will apply the **product rule** since this is the product of two functions: \( x^3 \) and \( f(x) \).

The **product rule** states:

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

Here, let:
- \( u(x) = x^3 \)
- \( v(x) = f(x) \)

Now, differentiate each term:
- \( u'(x) = \frac{d}{dx} [x^3] = 3x^2 \)
- \( v'(x) = f'(x) \) (since we don't know the explicit form of \( f(x) \))

Now apply the product rule:

\[
h'(x) = u'(x)v(x) + u(x)v'(x)
\]

Substitute the terms:

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

Thus, the derivative of \( h(x) = x^3 f(x) \) is:

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

Would you like to work through specific examples of \( f

Learn how to differentiate h(x) = x^3 f(x) using the product rule. This solution involves key calculus concepts, including derivatives and the product rule theorem. Suitable for students in Grades 11-12 or early university levels.

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