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Implicit Differentiation

Implicit Differentiation. Objective: To find derivatives of functions that we cannot solve for y. Implicit Differentiation. 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 (but often it is easier to do so).

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Implicit Differentiation

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  1. Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.

  2. Implicit Differentiation • 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 (but often it is easier to do so). • Find dy/dx for . Can we solve this for y?

  3. Implicit Differentiation • 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. • For example, we can take the derivative of with the quotient rule:

  4. Implicit Differentiation • We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.

  5. Example 2 • Use implicit differentiation to find dy/dx if

  6. Example 2 • Use implicit differentiation to find dy/dx if

  7. Example 2 • Use implicit differentiation to find dy/dx if

  8. Example 2 • Use implicit differentiation to find dy/dx if

  9. Example 3 • Use implicit differentiation to find if

  10. Example 3 • Use implicit differentiation to find if

  11. Example 3 • Use implicit differentiation to find if

  12. Example 3 • Use implicit differentiation to find if

  13. Example 3 • Use implicit differentiation to find if

  14. Example 3 • Use implicit differentiation to find if

  15. Example 3 • Use implicit differentiation to find if

  16. Example 4 • Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).

  17. Example 4 • Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). • We know that the slope of the tangent line means the value of the derivative at the given points.

  18. Example 4 • Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). • We know that the slope of the tangent line means the value of the derivative at the given points.

  19. Example 4 • Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1). • We know that the slope of the tangent line means the value of the derivative at the given points.

  20. Example 5 Use implicit differentiation to find dy/dx for the equation .

  21. Homework • Page 241-242 • 1-23 odd • 27 (just use implicit)

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