what is the derivative of the product of two functions

The derivative of the product of two functions \( f(x) \) and \( g(x) \) can be found using the **product rule**. The product rule states:

\[
\frac{d}{dx} [f(x) \cdot g(x)] = f(x) \cdot g'(x) + g(x) \cdot f'(x)
\]

In words, the derivative of the product of two functions is:

1. The first function multiplied by the derivative of the second function
2. Plus the second function multiplied by the derivative of the first function

So, if you have two differentiable functions \( f(x) \) and \( g(x) \), the derivative of their product \( f(x) \cdot g(x) \) is:

\[
f(x) \cdot g'(x) + g(x) \cdot f'(x)
\]

### Example
If \( f(x) = x^2 \) and \( g(x) = \sin(x) \), then:

1. \( f'(x) = 2x \)
2. \( g'(x) = \cos(x) \)

Using the product rule:

\[
\frac{d}{dx} [x^2 \cdot \sin(x)] = x^2 \cdot \cos(x) + \sin(x) \cdot 2x
\]

Would you like further details on applying the product rule or have any other questions?

Here are five follow-up questions that might help deepen your understanding:

1. How do you apply the product rule to more than two functions?
2. What is the relationship between the product rule and the chain rule?
3. How would the product rule apply if one of the functions is a constant?
4. What happens when both functions are exponential functions?
5. Can you apply the product rule in implicit differentiation?

**Tip:** The product rule is essential in calculus, especially when differentiating complex expressions involving multiple functions. It can often be combined with other rules, like the chain rule, for even more complex derivatives.

Learn to find the derivative of the product of two functions using the product rule in calculus. This guide provides a detailed explanation, example, and follow-up questions to deepen understanding. Suitable for high school calculus students in Grades 11-12.

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