Implicit differentiation
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Implicit Differentiation. Lesson 3.5. Introduction. Consider an equation involving both x and y: This equation implicitly defines a function in x It could be defined ex plicitly. Differentiate. Differentiate both sides of the equation each term one at a time

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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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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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Substitute

Second Derivative

  • Given x2 –y2 = 49

  • y’ =??

  • y’’ =


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Note: this is a constant

Exponential & Log Functions

  • Given y = bx where b > 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 TI’salready havethis function


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Assignment

  • Lesson 3.5

  • Page 171

  • Exercises 1 – 81 EOO


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