### DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS

None of the six basic trigonometry functions is a one-to-one function. However, in the following list, each trigonometry function is listed with an appropriately restricted domain, which makes it one-to-one.
1. for
2. for
3. for
4. for , except
5. for , except x = 0
6. for
Because each of the above-listed functions is one-to-one, each has an inverse function. The corresponding inverse functions are
1. for
2. for
3. for
4. arc for , except
5. arc for , except y = 0
6. arc for
In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . The derivatives of the above-mentioned inverse trigonometric functions follow from trigonometry identities, implicit differentiation, and the chain rule. They are as follows.
1. arc
2. arc
3. arc
In the list of problems which follows, most problems are average and a few are somewhat challenging.

• PROBLEM 1 : Differentiate .

• PROBLEM 2 : Differentiate .

• PROBLEM 3 : Differentiate arc arc .

• PROBLEM 4 : Let arc . Solve f'(x) = 0 for x .

• PROBLEM 5 : Let . Show that f'(x) = 0 . Conclude that .

• PROBLEM 6 : Evaluate .

Some of the following problems require use of the chain rule.

• PROBLEM 7 : Differentiate .

• PROBLEM 8 : Differentiate .

• PROBLEM 9 : Differentiate .

• PROBLEM 10 : Determine the slope of the line tangent to the graph of at x = e .

• PROBLEM 11 : Differentiate arc . What conclusion can be drawn from your answer about function y ? What conclusion can be drawn about functions arc and ?

• PROBLEM 12 : Differentiate .

• PROBLEM 13 : Find an equation of the line tangent to the graph of at x=2 .

• PROBLEM 14 : Evaluate .

• PROBLEM 15 : A movie screen on the front wall in your classroom is 16 feet high and positioned 9 feet above your eye-level. How far away from the front of the room should you sit in order to have the ``best" view ? (HINT: Find the largest possible angle in the given diagram below.)