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Notes lesson 2.5 Implicit Differentiation

NotebookTable of content

Page1

Learning Target 1

6) 2.1 The Derivative

7)2.2 The power rule

8) 2.3 The Product Rule/quotient rule

9)2.4 chain rule

10) Differentiability

11) Implicit differentiation

LT1: Find the derivative of a function defined implicitly

HW: p 146; 1,5,9,11,15,25,26,28,29,45,47,51,57,65,69

2.5 Implicit Differentiation

Slope – The derivative Type of equation3 1y x Explicit Equation

Question :Tangent line to equation at x =2 ?

23x'ydy

dxslope 12 1 1y y m x x

9 12 2y x

2 2 25x y Implicit :

1) Does it have slope ?

Yes, not at all places 2) What is the tangent line at x = 3 ?

2 2 25x y

2) What is the tangent line at x = 3 ?

2 2 ' 0x y y

2 ' 2y y x

2'2

xy

y

2 23 25y

(3,4)

4y

(3, 4)

'x

yy

3'4

y

34 3

4y x

3'

4y

34 3

4y x

3 22 3 7x y x

Find the derivative.

27 6'

6

xy

y

27 6

6

x

y

27 6

6

x

y

Other possible solutions

dy

dxSteps to implicit differentiate

1) Take the derivative with respect to x

2) Collect y’ to one side and all other stuff to other side.

3) Factor y’ 4) Divide by the part that does

not have y’ .

26 6 ' 7x y y

26 ' 6 7y y x

26 7'

6

xy

y

Find the derivative.4 3 5 72 13y x y x y 2 5 6

3 3 4

3 14'4 5 13

x y xy

y x y

6 2 5

3 3 4

14 3'4 5 13

x x yy

y x y

6 2 5

3 3 4

14 3'

4 5 13

x x yy

y x y

Other possible solutions

dy

dx

3 2 5 3 4 64 ' 3 5 ' 14 13 'y y x y x y y x y

3 3 4 2 5 64 ' 13 ' 5 ' 3 14y y y x y y x y x

3 3 4 2 5 6' 4 13 5 3 14y y x y x y x

2 5 6

3 3 4

3 14'4 13 5

x y xy

y x y

Find the derivative.

sin x y x y 'y

yx

'y

yx

Other possible solutions

dy

dx

cos 1 ' 'x y y x y y xy

cos ' 'x y y xy y xy

cos 'cos 'y x y xy x y y xy

'cos ' cosxy x y xy y y x y

' cos cosy x x y x y y x y

cos

'cos

y y x yy

x x y x

1 cos'

cos 1

y x yy

x x y

'y

yx

Find the 2nd derivative.

2 2 25x y

'x

yy

2

2

d y

dx

We already know the 1st derivative .

2

'''

y xyy

y

2

'''

y xyy

y

2

'

''

xy

y x

yy

Substitute the 1st derv. In.

2

2''

xy

yy

y

Simplify

2 2

2''

y xy y

yy

2 2

2''

y xy

yy

2 2

3''

y xy

y

Find the 2nd derivative.2 4 5x y x

2

4 2'

xyy

x

3

6 16''

xyy

x

2

2

d y

dx

Solution: