1. U2 L5 Quotient Rule QUOTIENT RULE
UNIT 2 LESSON 5
QUOTIENT RULE

2. If you thought the product rule was bad...
U2 L5 Quotient Rule
If you thought the product rule was bad...

3. Function f is the QUOTIENT of functions g and h
U2 L5 Quotient Rule
Function f is the QUOTIENT of functions g and h
It would be nice if the rule were
But, it is NOT!!!

4. Let’s show that if f(x) = x3 + x2 – 5x and g(x) = – x then
Example 1
Let’s show that if f(x) = x3 + x2 – 5x and g(x) = – x then
U2 L5 Quotient Rule
then =
and

5. If and and so ≠ From previous slide then Example 1 continued
U2 L5 Quotient Rule
Example 1 continued
If and
and so ≠
From previous slide
then

6. 𝒇 𝒈 ′ 𝒙 = 𝒈 𝒙 𝒇 ′ 𝒙 −𝒇 𝒙 𝒈′(𝒙) 𝒈(𝒙) 𝟐
U2 L5 Quotient Rule
Quotient Rule
In this section we develop a formula for the derivative of the quotient of two functions.
𝒇 𝒈 ′ 𝒙 = 𝒈 𝒙 𝒇 ′ 𝒙 −𝒇 𝒙 𝒈′(𝒙) 𝒈(𝒙) 𝟐

7. U2 L5 Quotient Rule
Let’s try it in English
The bottom (g (x)) times the derivative of the top f ′ (x)
minus the top (f (x)) times the derivative of the bottom g ′ (x)
all over the bottom g (x) squared.

8. Example 1 continued Quotient Rule x ≠ 0 FROM BEFORE if
U2 L5 Quotient Rule
Example 1 continued
FROM BEFORE if
𝒇 𝒙 = 𝒙 𝟑 + 𝒙 𝟐 −𝟓𝒙
𝒈 𝒙 =−𝒙
then
and
Quotient Rule
x ≠ 0

9. If f(x) = x2 + 3x + 2 and g(x) = x + 1 then
U2 L5 Quotient Rule
EXAMPLE 2
If f(x) = x2 + 3x and g(x) = x then

10. Use f(x) = x2 + 3x + 2 and g(x) = x + 1 to show that
Example 2 continued
U2 L5 Quotient Rule
Use f(x) = x2 + 3x and g(x) = x to show that ≠
but from previous slide

11. Use f(x) = x2 + 3x + 2 and g(x) = x + 1 to show that
Example 2 continued
U2 L5 Quotient Rule
Use f(x) = x2 + 3x + 2 and g(x) = x + 1 to show that
and from previous slide =

12. Example 3 Using the Quotient Rule
U2 L5 Quotient Rule
Differentiate State any restrictions on the domain.
Restriction on domain
x ≠ 4

13. Example 4 Using the Quotient Rule
U2 L5 Quotient Rule
Example 4 Using the Quotient Rule
Differentiate State any restrictions on the domain.
Since x2 + 1 is always > 0 there
are no restrictions

14. Example 5 Using the Quotient Rule
U2 L5 Quotient Rule
Example 5 Using the Quotient Rule
𝑦= 𝟐 𝒙 𝟑 +𝟓 𝟑𝒙
𝒅𝒚 𝒅𝒙 = 𝟑𝒙 𝟔 𝒙 𝟐 −(𝟐 𝒙 𝟑 +𝟓)(𝟑) 𝟑𝒙 𝟐
𝒅𝒚 𝒅𝒙 = 𝟏𝟖 𝒙 𝟑 −𝟔 𝒙 𝟑 −𝟏𝟓 𝟗 𝒙 𝟐
𝒅𝒚 𝒅𝒙 = 𝟏𝟐 𝒙 𝟑 −𝟏𝟓 𝟗 𝒙 𝟐 , 𝒙≠𝟎
𝒅𝒚 𝒅𝒙 = 𝟑(𝟒 𝒙 𝟑 −𝟓) 𝟗 𝒙 𝟐 = 𝟒 𝒙 𝟑 −𝟓 𝟑 𝒙 𝟐 , 𝒙≠𝟎

15. Example 6 Using the Quotient Rule
U2 L5 Quotient Rule
𝑦= 3−𝑥 6−5𝑥
Differentiate using the Quotient Rule.
𝒅𝒚 𝒅𝒙 = (𝟔−𝟓𝒙) −𝟏 −(𝟑−𝒙)(−𝟓) 𝟔−𝟓𝒙 𝟐
𝒅𝒚 𝒅𝒙 = −𝟔+𝟓𝒙+𝟏𝟓−𝟓𝒙 𝟔−𝟓𝒙 𝟐
𝒅𝒚 𝒅𝒙 = 𝟗 𝟔−𝟓𝒙 𝟐

16. Example 6 continued Find the slope of the tangent at P(0, ½ )
U2 L5 Quotient Rule
Example 6 continued
Find the slope of the tangent at P(0, ½ )
From previous slide
𝒅𝒚 𝒅𝒙 = 𝟗 𝟔−𝟓𝒙 𝟐
𝒅𝒚 𝒅𝒙 = 𝟗 𝟔−𝟓(𝟎 𝟐 ) = 𝟗 𝟔 = 𝟑 𝟐

17. Example 7 Application x = 0 or x = -5 At what points on the curve
U2 L5 Quotient Rule
Example 7 Application
At what points on the curve
is the tangent line horizontal?
The tangent line will be horizontal when the derivative = 0
2x(x + 5) = 0
x = 0 or x = -5
Points are (0, 0) and (-5, -5)

18. Example 7 Application y = 0 y = -5 (0, 0) (- 5, - 5)
U2 L5 Quotient Rule
Example 7 Application
(0, 0)
y = 0
y = -5
(- 5, - 5)

19. Complete Homework Assignment
Questions 1-5