1 / 14

Implicit Differentiation

Implicit Differentiation. Section 3.7a. Consider the equation:. Is this a function?. Is the equation differentiable?. If so, how do we differentiate?. We use implicit differentiation , so named b/c the functions are defined implicitly (hidden) within the original equation .

luna
Download Presentation

Implicit Differentiation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Implicit Differentiation Section 3.7a

  2. Consider the equation: Is this a function? Is the equation differentiable? If so, how do we differentiate?

  3. We use implicit differentiation, so named b/c the functions are defined implicitly (hidden) within the original equation

  4. Implicit Differentiation Process  We treat y as a differentiable function of x!!! • Differentiate both sides of the equation with • respect to x. Because of the Chain Rule, any time differentiating any term containing a y, also multiply by dy/dx!!! 2. Collect the terms with dy/dx on one side of the equation. 3. Factor out dy/dx. 4. Solve for dy/dx.

  5. Initial Guided Practice Find dy/dx: Implicit Differentiation: Solve for the derivative: Does this answer make sense graphically?

  6. Initial Guided Practice Find the slope of the circle at First, find the slope of any point on the circle via implicit differentiation: Slope at the given point: Again, verify this answer graphically!

  7. Initial Guided Practice Show that the slope dy/dx is defined at every point on the graph of Imp Diff: This formula for dy/dx is defined at every point (x,y) except for those points at which cos(y)=2… Which never happens!!!

  8. We can use all of this new info to expand the POWER of the POWER RULE!!!  A proof: First, let p and q be integers with q > 0 and suppose that: Subst. for y Law of Exp. Imp. Diff.! Final Answer!

  9. The EXPANDED Power Rule: If n is any rational number, then (If n < 1, then the derivative does not exist at x = 0) WHY NOT??? Note: Before, the power rule worked for integers only…  now, it works for any rational number power (i.e., fraction) What is the derivative of the square root function???

  10. Additional Guided Practice For each of the following, find dy/dx.

  11. Additional Guided Practice For each of the following, find dy/dx.

  12. Additional Guided Practice For each of the following, find dy/dx.

  13. Additional Guided Practice For each of the following, find dy/dx.

  14. Additional Guided Practice For each of the following, find dy/dx.

More Related