Lesson 3-2
Product and Quotient Rules
Objectives
• Use product and quotient rules of
differentiation
Vocabulary
• Function – an independent variable (x or t) yields
only one dependent variable value
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.
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.
Example 1
Find the derivatives of the following:
1. f(t) = (7t – 12) (4t3)
f’(t) = (7)(4t³) + (7t – 12)(12t²)
2. f(x) = 6(7x -3) (2x²)
f’(x) = 6(7)(2x²) + 6(7x – 3)(4x)
Example 2
Find the derivatives of the following:
3. f(x) = (2x4 + 3x2 + 7) (9 - x³)
f’(x) = (8x3 + 6x)(9 - x³) + (-3x²)(2x4 + 3x + 7)
4. y = -(6x³ + 5x² - 8x + 2) (4 – x)
y’(x) = -(18x2 + 10x - 8)(4 - x) + -(6x³ + 5x² - 8x + 2)(-1)
Example 3
Find the derivatives of the following:
5. d(t) = (4t) (10 – 4t)
d’(t) = (4)(10 – 4t) + (4t)(-4)
6. g(t) = (7t4 – 4t3) (6t2 + 9t – 19)
g’(t) = (28t3 – 12t2) (6t2 + 9t – 19) + (7t4 – 4t3) (12t + 9)
Example 4
Find the derivatives of the following:
7. y = (2 – 4x) / (x² – 3x³)
(x² - 3x³)(-4) – (2x - 9x²) (2 – 4x)
y’(x) = ---------------------------------------------(x² - 3x³)²
8. f(x) = (2 – 3x + 5x² – 8x³) / 9
(9)(– 3 + 10x – 24x²) – (0) (2 – 3x + 5x² – 8x³)
y’(x) = ---------------------------------------------------------------(9)²
Example 5
Find the derivatives of the following:
9. f(x) = ex / x
(ex)(1) – (ex) (x)
y’(x) = -----------------------(ex)²
10.f(x) = e2 / x4
(x4)(0) – (4x3) (e²)
y’(x) = --------------------------(x4)²
Example 6
Find the derivatives of the following:
11. f(t) = (6t + 2) / (7t - 1)
(7t – 1)(6) – (7)(6t + 2)
f’(t) = -------------------------------------(7t – 1)²
12. y = (2x + 1) / (3x + 4)
(3x + 4)(2) – (3)(2x + 1)
y’(x) = -------------------------------------(3x + 4)²
Summary & Homework
• Summary:
– 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 197-198: 3, 5-7, 9, 13, 14, 17, 31, 35