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Derivatives of Inverse Trigonometric Functions

Derivatives of Inverse Trigonometric Functions. Chapter 4.3. Derivatives of Inverse Functions. Derivatives of Inverse Functions. How are the slopes of the tangent lines from the previous slide related? The slope of the tangent line at for is

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Derivatives of Inverse Trigonometric Functions

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  1. Derivatives of Inverse Trigonometric Functions Chapter 4.3

  2. Derivatives of Inverse Functions

  3. Derivatives of Inverse Functions • How are the slopes of the tangent lines from the previous slide related? • The slope of the tangent line at for is • Since the tangent line at for is a reflection of the above tangent line, we can find its equation in terms of by swapping and , then solving for

  4. Derivatives of Inverse Functions • The slopes are reciprocals • This gives us a hint about the relationship between the derivative of a function and its inverse (when an inverse exists) • The next theorem asserts the existence of the derivative of inverse function (without proof) • In addition, the relationship is given and can be proven using the Chain Rule

  5. Derivatives of Inverse Functions THEOREM: If is differentiable at every point of an interval and is never zero on , then has an inverse and is differentiable at every point of the interval . Furthermore,

  6. Derivatives of Inverse Functions THEOREM: PROOF: We can easily prove the last part of this theorem using the Chain Rule. If , then Now, is an inner function. So we apply the Chain Rule to get

  7. Derivative of Arcsine • The function is differentiable on the open interval • Its derivative is positive (i.e., non-zero) in the same interval • By the previous theorem, there must exist a function that is differentiable on the interval • We can use implicit differentiation to find the derivative of arcsine

  8. Derivative of Arcsine Since is a function of , then is an implicit function This division presents no problems since the cosine is non-zero in the interval Since , then

  9. Derivative of Arcsine But , so More generally, if is a differentiable function of , then

  10. Example 1: Applying the Derivative Differentiate .

  11. Example 1: Applying the Derivative Differentiate . We have as an inner function, so if and , then

  12. Derivative of Arctangent • The domain of arctangent is • It is differentiable for all real numbers • We find the derivative of arctangent in the same way we found the derivative of arcsine Use implicit differentiation (since is a function of )

  13. Derivative of Arctangent Use the Pythagorean identity : But so we get

  14. Derivative of Arctangent • Note that this is defined for all • More generally, if is a function of , then

  15. Example 2: A Moving Particle A particle moves along the -axis so that its position at any time is given by . What is the velocity of the particle when ?

  16. Example 2: A Moving Particle A particle moves along the -axis so that its position at any time is given by . What is the velocity of the particle when ? We have as an inner function, so if and , then

  17. Example 2: A Moving Particle A particle moves along the -axis so that its position at any time is given by . What is the velocity of the particle when ? At the velocity is

  18. Derivative of Arcsecant • We must be careful to follow the requirements of the theorem on derivatives of inverses when determining the derivative of arcsecant • These are • must be differentiable over its domain, • must not equal zero in this interval • Note from the graph of that the derivatives at both and are horizontal lines • So the interval we will choose will be

  19. Derivative of Arcsecant

  20. Derivative of Arcsecant • Having set up the conditions required by the theorem, we now conclude that has an inverse that that the inverse is differentiable over , or • Now, if , then • Differentiate this implicitly • Note that the denominator will not be zero because of the domain we have chosen

  21. Derivative of Arcsecant • It remains to rewrite this as a function of • Since and , then • How do we deal with the ?

  22. Derivative of Arcsecant • Recall that the domain of is , or • We can rewrite the derivative as a piecewise function

  23. Derivative of Arcsecant This is the same as

  24. Derivative of Arcsecant Hence, If is a function of , then

  25. Example 3: Finding the Derivative of Arcsecant Differentiate .

  26. Example 3: Finding the Derivative of Arcsecant Differentiate . is an inner function. Using the Chain Rule we have and Therefore,

  27. Derivatives of the Other Three Inverses • We could use the same procedure to find the derivatives of arccosine, arccotangent, and arccosecant • But it is easier to use identities; the one below shows how arccosine and arcsine are related Therefore,

  28. Derivatives of the Other Three Inverses • The other two are similarly derived • They are

  29. Derivatives of the Other Three Inverses • The derivative of arccosecant is

  30. Derivatives of the Other Three Inverses • All six derivatives are given below

  31. Calculator Conversion Identities • Calculators have keys for arcsine, arccosine, and arctangent • The following should be used to calculate arccosecant, arcsecant, and arccotangent

  32. Example 4: A Tangent Line to the Arccotangent Curve Find an equation for the line tangent to the graph of at .

  33. Example 4: A Tangent Line to the Arccotangent Curve Find an equation for the line tangent to the graph of at . The derivative of is The slope of the tangent line at is The tangent line equation is

  34. Exercise 4.3

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