Quotient Rule

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

if y = u/v, then y' = (v * u' - u * v') / v^2

Example:

Find the derivative of the function y = (3x^2 + 2x) / (x + 1)

First, identify u and v:

u = 3x^2 + 2x

v = x + 1

Next, find the derivatives of u and v:

u' = 6x + 2

v' = 1

Now, apply the quotient rule formula:

y' = ((x + 1)(6x + 2) - (3x^2 + 2x)(1)) / (x + 1)^2

Simplify the expression to get the derivative of y:

y' = (6x^2 + 2x + 6x + 2 - 3x^2 - 2x) / (x + 1)^2

y' = (3x^2 + 6x + 2) / (x + 1)^2

Practice Problems:

1. Find the derivative of y = (4x^3 - 2x^2) / (2x + 1)
2. Find the derivative of y = (x^2 + 3x + 1) / (x - 2)
3. Find the derivative of y = (2x^4 - 5x^2 + 1) / (3x + 2)

Remember to simplify your final answer as much as possible!

Hope this helps!