Implicit Differentiation

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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

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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

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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

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

• Substitute x 3, y -2

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Second Derivative

• Given x2 y2 49
• y ??
• y

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Exponential Log Functions

• Given y bx where b gt 0, a constant
• Given y logbx

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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

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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 TIsalready havethis function
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Assignment

• Lesson 3.5
• Page 171
• Exercises 1 81 EOO