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3.6 - Implicit Differentiation (page 211-217)

3.6 - Implicit Differentiation (page 211-217). We have been differentiating functions that are expressed in the form y=f(x). An equation in this form is said to define y explicitly as a function of x.

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3.6 - Implicit Differentiation (page 211-217)

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  1. 3.6 - Implicit Differentiation(page 211-217) • We have been differentiating functions that are expressed in the form y=f(x). • An equation in this form is said to define yexplicitly as a function of x. • The equations on the next slide are not defined explicitly although, the first two may be manipulated to be written as functions in the form y=f(x) the second two equation are not functions.

  2. Implicit Equations

  3. Implicit Equations(page 212-213) • In the original equations 1,2,3 on the previous slide we could say that the equation defines yimplicitly as a function(s) of x. • The 4th equation, called the“Folium of Descartes”, could not be written explicitly to define y in terms of x. • It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation.

  4. Implicit Differentiation • We differentiate both sides of the equation and treat y as a composite function and apply the chain rule when necessary. • This method of obtaining derivatives is called implicit differentiation. • We will use this technique to solve word problems like the one on the next slide involving “Related Rates”

  5. Related Rate Problem (page 220)

  6. Example 1(page 213)

  7. Example 1(page 248)

  8. Example 2(page 214)

  9. Example 3(page 214)

  10. Example 3(page 214)

  11. Example 3(page 214)

  12. Example 4(page 215)

  13. Example 4(page 215)

  14. Example 4(page 215) See graph on next slide.

  15. Example 4(page 215)

  16. Example 4(page 215) See graph on next slide

  17. Example 4(page 215)

  18. Differentiability of Functions Defined Implicitly(page 216) • When differentiating implicitly, it is assumed that y represents a differentiable function of x. • If function is not differentiable, then the derivative will be meaningless. • We will not be determining whether or not an implicitly defined function is differentiable. “We will leave such matters for more advanced courses.

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