1 6.5 Derivatives of Inverse Trigonometric Functions Approach: to differentiate an inverse trigonometric function, we reduce the expression to trigonometric functions and use the rule of implicit differentiation. Example: Differentiate the arcsine function: Solution: This same equality can be rewritten as We need to find dy/dx, which can be done implicitly: Next, we want to write this derivative as a function of x, not y:

2 Example (cntd): To reduce the derivative to the function of x, we use the fact that sin y=x, and the trigonometric identity that gives We use the fact that the range of the arcsine function is restricted to Since the cosine function takes only positive values in this interval, the positive sign must be chosen: Using this equality, we write the derivative in the final form:

3 Exercise: Differentiate the arccosine function: Example: Differentiate the arctangent function Solution: Differentiate implicitly: Apply some trigonometry to write the result as a function of x: Finally:

4 Generalized inverse trigonometric rules (using chain rule) Inverse trigonometric rules

5 Exercises: Differentiate

6 Homework Section 6.5: 1,7,19,25,27,31,37,41.