Product Rule in Calculus

The product rule is a formula used to find the derivative of the product of two functions. It is often used when finding the derivative of a function that can be expressed as the product of two other functions.

Formula

The product rule states that if u and v are functions of x, then the derivative of their product uv is given by:

(uv)' = u'v + uv'

Where u' and v' are the derivatives of u and v with respect to x, respectively.

Example

Let's say we have two functions u(x) = 3x² and v(x) = 2x³. To find the derivative of their product uv, we can use the product rule:

u'(x) = 6x, v'(x) = 6x²

So, (uv)' = (3x²)'(2x³) + 3x²(2x³)'

= 6x(2x³) + 3x²(6x²)

= 12x⁴ + 18x⁴

= 30x⁴

Study Guide

Understand the product rule formula: (uv)' = u'v + uv'
Practice finding the derivatives of products of functions using the product rule.
Be familiar with differentiating various types of functions that can be expressed as products.
Use the product rule to solve problems involving rates of change, optimization, and related rates.