1. Product and Quotient Rules
Lesson 3-2
Product and Quotient Rules

2. Objectives
Use product and quotient rules of differentiation

3. Vocabulary
Function – an independent variable (x or t) yields only one dependent variable value

4. Product Differentiation Rule
d d d
---- [f(x) • g(x)] = f(x) • g(x) + g(x) • f(x)
dx dx dx
In words: the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

5. Quotient Differentiation Rule
d d
g(x) [f(x)] – f(x) -----[g(x)]
d f(x) dx dx
---- [ ] =
dx g(x) [g(x)]²
In words: the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

6. Example 1 Find the derivatives of the following:
f(t) = (7t – 12) (4t3)
f(x) = 6(7x -3) (2x²)
f’(t) = (7)(4t³) + (7t – 12)(12t²)
f’(x) = 6(7)(2x²) + 6(7x – 3)(4x)

7. Example 2 Find the derivatives of the following:
f(x) = (2x4 + 3x2 + 7) (9 - x³)
y = -(6x³ + 5x² - 8x + 2) (4 – x)
f’(x) = (8x3 + 6x)(9 - x³) + (-3x²)(2x4 + 3x + 7)
y’(x) = -(18x2 + 10x - 8)(4 - x) + -(6x³ + 5x² - 8x + 2)(-1)

8. Example 3 Find the derivatives of the following: d(t) = (4t) (10 – 4t)
g(t) = (7t4 – 4t3) (6t2 + 9t – 19)
d’(t) = (4)(10 – 4t) + (4t)(-4)
g’(t) = (28t3 – 12t2) (6t2 + 9t – 19) + (7t4 – 4t3) (12t + 9)

9. Example 4 Find the derivatives of the following:
y = (2 – 4x) / (x² – 3x³)
f(x) = (2 – 3x + 5x² – 8x³) / 9
(x² - 3x³)(-4) – (2x - 9x²) (2 – 4x)
y’(x) =
(x² - 3x³)²
(9)(– x – 24x²) – (0) (2 – 3x + 5x² – 8x³)
y’(x) =
(9)²

10. Example 5 Find the derivatives of the following: f(x) = ex / x
(ex)(1) – (ex) (x)
y’(x) =
(ex)²
(x4)(0) – (4x3) (e²)
y’(x) =
(x4)²

11. Example 6 Find the derivatives of the following:
f(t) = (6t + 2) / (7t - 1)
y = (2x + 1) / (3x + 4)
(7t – 1)(6) – (7)(6t + 2)
f’(t) =
(7t – 1)²
(3x + 4)(2) – (3)(2x + 1)
y’(x) =
(3x + 4)²

12. Summary & Homework Summary: Homework:
Product rule allows us to find the derivative of functions multiplied together
Quotient rule allows us to find the derivative of functions divided
Remember a denominator is (den(x))-1
Homework:
pg : 3, 5-7, 9, 13, 14, 17, 31, 35