Suppose f(x) = g(x)h(x).
(a) What is the formula for f
0
(x)?
(b) Use this formula to differentiate f(x) = (2x
3 + 4)(6 − x
4
)


### (a) Formula for \( f'(x) \):

If \( f(x) = g(x)h(x) \), then the derivative of \( f(x) \), using the product rule, is:

\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]

### (b) Differentiating \( f(x) = (2x^3 + 4)(6 - x^4) \):

Here, we have:

- \( g(x) = 2x^3 + 4 \)
- \( h(x) = 6 - x^4 \)

We need to find \( g'(x) \) and \( h'(x) \) first:

- \( g'(x) = \frac{d}{dx}(2x^3 + 4) = 6x^2 \)
- \( h'(x) = \frac{d}{dx}(6 - x^4) = -4x^3 \)

Now, using the product rule formula:

\[
f'(x) = g'(x)h(x) + g(x)h'(x)
\]

Substituting \( g(x), g'(x), h(x), \) and \( h'(x) \):

\[
f'(x) = (6x^2)(6 - x^4) + (2x^3 + 4)(-4x^3)
\]

Now, let's expand this expression:

\[
f'(x) = 6x^2(6 - x^4) + (2x^3 + 4)(-4x^3)
\]

\[
f'(x) = 36x^2 - 6x^6 - 8x^6 - 16x^3
\]

Simplifying the terms:

\[
f'(x) = 36x^2 - 14x^6 - 16x^3
\]

Thus, the derivative is:

\[
f'(x) = -14x^6 - 16x^3 + 36x^2
\]

---

Would you like any further details or explanations? Here are 5 questions to explore further:

1. How does the product rule differ from the chain rule in calculus?
2. What would the derivative of \( f(x) \) look like if both \( g(x) \) and \( h(x) \) were more complex?
3. What happens to the derivative if one of the functions in the product is a constant?
4. Can the product rule be extended to the product of more than two functions?
5. How does the second derivative \( f''(x) \) change compared to \( f'(x) \)?

**Tip:** When using the product rule, always carefully compute the derivatives of each individual function before applying the rule to avoid mistakes.

Suppose f(x) = g(x)h(x).
(a) What is the formula for f'(x)?
(b) Use this formula to differentiate f(x) = (2x^3 + 4)(6 − x^4)

Discover how to apply the product rule to differentiate the function f(x) = (2x^3 + 4)(6 − x^4). This problem covers the product rule in calculus, polynomial differentiation, and step-by-step expansion. Perfect for students in Grades 11-12 studying calculus.

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