How does the product rule apply to calculate derivatives?

The product rule is a fundamental concept in calculus that allows you to find the derivative of the product of two or more functions. It's a crucial tool for calculating derivatives when you can't easily differentiate the product directly. Here's how it works:

1. Product Rule Formula: The product rule states that if you have two functions, f(x) and g(x), then the derivative of their product (f(x) * g(x)) is given by:

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

2. Application: To apply the product rule, follow these steps:

   • Identify the two functions, f(x) and g(x), in the given expression.
   • Differentiate each function with respect to x. This gives you $\frac{dg}{dx}$ and $\frac{df}{dx}$.
   • Multiply the first function by the derivative of the second, and the second function by the derivative of the first.
   • Add these two results together to get the derivative of the product.

Here's an example to illustrate the process:

• Consider the function $h(x) = (3x^2)(x^3)$.
• Identify $f(x) = 3x^2$ and $g(x) = x^3$.
• Differentiate each function: $\frac{dg}{dx} = 3x^2$ and $\frac{df}{dx} = 6x$.
• Apply the product rule: $h'(x) = f(x) \frac{dg}{dx} + g(x) \frac{df}{dx} = (3x^2)(3x^2) + (x^3)(6x) = 9x^4 + 6x^4 = 15x^4$.

So, the derivative of $h(x) = (3x^2)(x^3)$ is $h'(x) = 15x^4$.