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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.

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Derivatives of the inverse trigonometric functions

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  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:

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