Quotient Rule

The quotient rule is a formula used to find the derivative of a function that is the quotient of two other functions.

Formula

The quotient rule is given by:

f'(x) = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2

Where:

• f(x) is the numerator function
• g(x) is the denominator function
• f'(x) is the derivative of the numerator function
• g'(x) is the derivative of the denominator function

Example

Let's find the derivative of the function f(x) = (2x + 1) / (x^2 + 3x + 2).

Using the quotient rule, we have:

f'(x) = ((x^2 + 3x + 2)*(2) - (2x + 1)*(2x + 3)) / (x^2 + 3x + 2)^2

After simplifying, we get the derivative:

f'(x) = (2x^2 + 6x + 4 - 4x^2 - 6x - 2) / (x^2 + 3x + 2)^2

f'(x) = (-2x^2 - 2) / (x^2 + 3x + 2)^2

Study Guide

To use the quotient rule, follow these steps:

1. Identify the numerator function f(x) and the denominator function g(x).
2. Find the derivatives f'(x) and g'(x) of the numerator and denominator functions, respectively.
3. Apply the quotient rule formula to find the derivative f'(x) of the original function.

Remember to always simplify your final answer if possible, and be careful with the signs when using the quotient rule.