Understanding Derivatives of Inverse Trigonometric Functions

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