This document provides instruction on the product rule and quotient rule for differentiation. The product rule states that the derivative of uv is u'v + v'u, where you take the derivative of the first function and multiply by the second, and vice versa, then add the results. The quotient rule states that the derivative of u/v is (u'v - uv')/v^2, where you take the derivative of the numerator and multiply by the original denominator, subtract the original numerator multiplied by the derivative of the denominator, and divide the whole expression by the denominator squared. Examples are provided to demonstrate applying each rule to differentiate expressions. Homework problems are assigned from the textbook for the next

1. Product & Quotient
Rule!
Calculus AB
Miss Cain

2. The Product
Rule
d/dx (uv) = u’v + v’u
Derive the u and
multiply it by the
original v. Then add that
to the derivative of v
times the original u

3. Example 1:
Find f ’(x) if f(x)=(x2+1)(x3+3)
f ‘ (x)= 2x(x3+3) + 3x2(x2+1)
f ‘ (x)= 2x4+6x+3x4+3x2
f ‘ (x)= 5x4+3x2+6x
u = (x2+1) v = (x3+3)
u’ = 2x v’ = 3x2

4. The Quotient
Ruled/dx (u/v) = (u’v) – (uv’) / (v2)
**Note the order DOES matter! You
must hold the bottom and derive
the top first.**
Derive the u and multiply it by the
original v. Then subtract the
original u multiplied by the
derivative of v. After subtracting
them, divide the difference by the

5. Example 2:
Differentiate f(x) = (x2-1) / (x2+1)
f ‘(x) = [2x(x2+1) – 2x(x2-1)] / (x2+1)(x2+1)
f ‘(x) = (2x3 + 2x – 2x3 + 2x) / (x2+1)2
f ‘(x) = (4x) / (x2+1)2
u = (x2-1) v = (x2+1)
u’ = 2x v’ = 2x

6. HOMEWORK!!!
Page 249 1-15 & 25-45
odds only
DUE THURSDAY!
Be prepared to share
your answers AND your
work in class