The Quotient RuleThe Quotient Rule

ObjectiveObjective

To use the quotient rule for To use the quotient rule for differentiation.differentiation.

ES: Explicitly assessing information ES: Explicitly assessing information and drawing conclusionsand drawing conclusions

The Product RuleThe Product Rule2(3 2 ) (5 4 )y x x x

dy

dxDoesDoes (3 4 ) x (4) ?? NO!

Take each derivative

The Quotient RuleThe Quotient Rule23 2

5 4

x xy

x

3 4

4

dy x

dx

Does ? NO

The derivative of a quotient The derivative of a quotient is not necessarily equal to is not necessarily equal to

the quotient of the the quotient of the derivatives.derivatives.

The Quotient RuleThe Quotient Rule

The Quotient RuleThe Quotient Rule

The derivative of a quotient must by The derivative of a quotient must by calculated using the quotient rule:calculated using the quotient rule:

2

( ) '( ) ( ) '( )then

[ ( )]

dy g x f x f x g x

dx g x

( )If

( )

f xy

g x

Low d High minusHigh d Low, allover

Low Low(low squared)

The Quotient RuleThe Quotient Rule

1.1. Imagine that the function is actually Imagine that the function is actually broken into 2 pieces, high and low.broken into 2 pieces, high and low.

23 2

5 4

x xy

x

23 2

5 4

x xy

x

The Quotient RuleThe Quotient Rule

2. In the numerator of a fraction, leave 2. In the numerator of a fraction, leave low piece alone and derive high low piece alone and derive high piece.piece.

dy

dx (5 4 )x (3 4 )x

The Quotient RuleThe Quotient Rule

3. Subtract: Leave high piece alone 3. Subtract: Leave high piece alone and derive low piece.and derive low piece.

dy

dx (5 4 )x (3 4 )x

23 2

5 4

x xy

x

2(3 2 )x x (4)

The Quotient RuleThe Quotient Rule

4. In the denominator: Square low 4. In the denominator: Square low piece.piece.

dy

dx (5 4 )x (3 4 )x

23 2

5 4

x xy

x

2(3 2 )x x (4)2(5 4 )x

This is the derivative!

2(3 2 )x x

The Quotient RuleThe Quotient Rule

dy

dx (5 4 )x (3 4 )x (4)

2(5 4 )x

dy

dx 15 12x 12x 28x

2(5 4 )x20x 216x

dy

dx

28x2(5 4 )x

20x 15

Final Answer

The Quotient RuleThe Quotient Rule

Low d High minusHigh d Low, allover

Low Low(low squared)

Final Answer

2(2 3)x ( 1)x

Example A: Find the derivativeExample A: Find the derivative1

2 3

xy

x

dy

dx (2 3)x (1) (2)

dy

dx 2 3x 2x 2

2(2 3)x

2

5

(2 3)

dy

dx x

Low d High minusHigh d Low, allover

Low Low(low squared)

Final Answer

2(2 4 3)x x

Example B: Find the derivativeExample B: Find the derivative

dy

dx (2 3 )x (4 4)x ( 3)

2(2 3 )x

dy

dx 8x 8 212x 12x 26x

2(2 3 )x12x 9

dy

dx

26x 8x 12(2 3 )x

22 4 3

2 3

x xy

x

Low d High minusHigh d Low, allover

Low Low(low squared)

2( 1)x (5) (5 2)x 2 2( 1)x

2

5 2

1

xy

x

Example C: Find the derivativeExample C: Find the derivative

dy

dx

dy

dx

25x 5 210x 4x2 2( 1)x

dy

dx

25x 4x 52 2( 1)x

(2 )x

Final Answer

Low d High minusHigh d Low, allover

Low Low(low squared)

2(5 2)x 2(12 7)x 3(4 7 )x x (10 )x2 2(5 2)x

Example D: Find the derivativeExample D: Find the derivative

dy

dx

dy

dx

460x 235x 224x 14 440x2 2(5 2)x

270x

dy

dx

420x 259x 142 2(5 2)x

3

2

4 7

5 2

x xy

x

Final Answer

Low d High minusHigh d Low, allover

Low Low(low squared)

2(5 4)x

(5 4)x [(1 2 )x (3) (3 2)x ( 2)] (1 2 )(3 2)x x (5)

Example E: Find the derivativeExample E: Find the derivative(1 2 )(3 2)

5 4

x xy

x

dy

dx

dy

dx

(5 4)x (3 6x2(5 4)x

6 4)x (3x 2 26x 4 )x (5)

dy

dx

(5 4)x ( 12 1)x 2(5 4)x

2(2 6 )x x (5)

Low d High minusHigh d Low, allover

Low Low(low squared)

Product Rule for D’Hi

The Quotient RuleThe Quotient Ruledy

dx

(5 4)x ( 12 1)x 2(5 4)x

2(2 6 )x x (5)

dy

dx

260x 5x2(5 4)x

48x 4 10 5x 230x

dy

dx

230x2(5 4)x

48x 6

Final Answer

The Quotient RuleThe Quotient Rule

Remember: The derivative of a Remember: The derivative of a quotient is quotient is Low, D-High, minus High, D-Low, all over the bottom squared.