How to find the derivative of a function using the product rule?

Product Rule for Derivatives

Now that you're familiar with basic derivatives and the sum rule, let's learn how to find the derivative of a function that's a product of two functions. The product rule is a crucial tool in calculus for this purpose. Here's how it works:

Given two functions, $f(x)$ and $g(x)$, and their product $h(x) = f(x)g(x)$, the derivative of $h(x)$ with respect to $x$ is given by:

$$\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$$

This is the product rule in calculus. It's often remembered by the mnemonic LEGS (which stands for Like Eating Grits Slowly).

Here's how to apply the product rule:

1. Identify $f(x)$ and $g(x)$ in the given function.
2. Find the derivatives of both functions, $f'(x)$ and $g'(x)$.
3. Apply the product rule formula: $f'(x)g(x) + f(x)g'(x)$.