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Implicit Differentiation

Lesson 3.5

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Introduction

• Consider an equation involving both x and y:

• This equation implicitly defines a function in x

• It could be defined explicitly

2 2 49x y

2 49 ( 7)y x where x

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Differentiate

• Differentiate both sides of the equation– each term– one at a time– use the chain rule for terms containing y

• For we get

• Now solve for dy/dx

2 2 49x y

2 2 0dy

x ydx

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Differentiate• Then gives us

• We can replace the y in the results with the explicit value of y as needed

• This gives usthe slope on the curve for any legal value of x

2 2 0dy

x ydx

2

2

dy x x

dx y y

2 49

dy x

dx x

View Spreadsheet Example

View Spreadsheet Example

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Guidelines for Implicit Differentiation

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Slope of a Tangent Line

• Given x3 + y3 = y + 21find the slope of the tangent at (3,-2)

• 3x2 +3y2y’ = y’

• Solve for y’2

2

3'1 3

xy

y

Substitute x = 3, y = -2 27

11slope

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Second Derivative

• Given x2 –y2 = 49

• y’ =??

• y’’ =

'x

yy

2

2 2

'd y y x y

dx y

Substitute

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Exponential & Log Functions

• Given y = bx where b > 0, a constant

• Given y = logbx

ln xdyb b

dx

1'ln

yb x

Note: this is a constant

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Using Logarithmic Differentiation

• Given

• Take the log of both sides, simplify

• Now differentiate both sides with respect to x, solve for dy/dx

10 3 7 818 ( 1) ( 3)y x x

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Implicit Differentiation on the TI Calculator

• On older TI calculators, you can declare a function which will do implicit differentiation:

• Usage:

Newer TI’salready havethis function

Newer TI’salready havethis function

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Assignment

• Lesson 3.5

• Page 171

• Exercises 1 – 81 EOO