1 / 16

Derivatives of the inverse trigonometric functions

Section 3.8. Derivatives of the inverse trigonometric functions. Derivatives of Inverse Functions. Theorem: If is differentiable at every point of an interval I and is never zero on I , then has an inverse and is differentiable at every point of the interval I.

tamas
Download Presentation

Derivatives of the inverse trigonometric functions

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. Section 3.8 Derivatives of the inverse trigonometric functions

  2. Derivatives of Inverse Functions Theorem: If is differentiable at every point of an interval I and is never zero on I, then has an inverse and is differentiable at every point of the interval I.

  3. Derivatives of Inverse Functions y x The slopes of inverse functions are reciprocals, at the corresponding points… in math symbols

  4. Derivatives of Inverse Functions Let . Given that the point is on the graph of , find the slope of the inverse of at . Our new rule: The slope of at is the reciprocal of the slope of at .

  5. Derivative of the Arcsine First, recall the graph: y –1 1 x So, should this function be differentiable across its entire domain??? • Everywhere except • at x = –1 or 1

  6. Derivative of the Arcsine

  7. Derivative of the Arcsine If is a differentiable function of with , applying the Chain Rule:

  8. Derivative of the Arctangent

  9. Derivative of the Arctangent If is a differentiable function of , again using the Chain Rule form:

  10. Derivative of the Arcsecant

  11. Derivative of the Arcsecant If is a differentiable function of with , and “chaining” once again, we have:

  12. Derivative of the Others Inverse Function – Inverse Cofunction Identities: • The derivatives of the inverse cofunctions • are the opposites (negatives) of the derivatives • of the corresponding inverse functions

  13. Guided Practice Find if

  14. Guided Practice Find if

  15. Guided Practice A particle moves along the x-axis so that its position at any time is . What is the velocity of the particle when ? First, find the general equation for velocity:

  16. Guided Practice A particle moves along the x-axis so that its position at any time is . What is the velocity of the particle when ? Now, at the particular time:

More Related