### SOLUTIONS TO DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS

SOLUTION 1 : Differentiate . Apply the product rule. Then  (Factor an x from each term.) .

SOLUTION 2 : Differentiate . Apply the quotient rule. Then     .

SOLUTION 3 : Differentiate arc arc . Apply the product rule. Then arc arc arc arc  arc arc = ( arc arc .

SOLUTION 4 : Let arc . Solve f'(x) = 0 for x . Begin by differentiating f . Then  (Get a common denominator and subtract fractions.)     .

(It is a fact that if , then A = 0 .) Thus,

2(x - 2)(x+2) = 0 .

(It is a fact that if AB = 0 , then A = 0 or B=0 .) It follows that

x-2 = 0 or x+2 = 0 ,

that is, the only solutions to f'(x) = 0 are

x = 2 or x = -2 .

SOLUTION 5 : Let . Show that f'(x) = 0 . Conclude that . Begin by differentiating f . Then .

If f'(x) = 0 for all admissable values of x , then f must be a constant function, i.e., for all admissable values of x ,

i.e., for all admissable values of x .

In particular, if x = 0 , then i.e., .

Thus, and for all admissable values of x .

SOLUTION 6 : Evaluate . It may not be obvious, but this problem can be viewed as a derivative problem. Recall that (Since h approaches 0 from either side of 0, h can be either a positve or a negative number. In addition, is equivalent to . This explains the following equivalent variations in the limit definition of the derivative.)  .

If , then , and letting , it follows that         .

The following problems require use of the chain rule.

SOLUTION 7 : Differentiate . Use the product rule first. Then (Apply the chain rule in the first summand.)  (Factor out . Then get a common denominator and add.)   .

SOLUTION 8 : Differentiate . Apply the chain rule twice. Then  (Recall that .)  .

SOLUTION 9 : Differentiate . Apply the chain rule twice. Then (Recall that .)   .

SOLUTION 10 : Determine the equation of the line tangent to the graph of at x = e . If x = e , then , so that the line passes through the point . The slope of the tangent line follows from the derivative (Apply the chain rule.)   .

The slope of the line tangent to the graph at x = e is   .

Thus, an equation of the tangent line is .