Lesson 3-8: Derivatives of Inverse Trig Functions AP Calculus Mrs. Mongold

At x = 2: We can find the inverse function as follows: To find the derivative of the inverse function: Switch x and y.

Slopes are reciprocals. At x = 2: At x = 4:

Slopes are reciprocals. Because x and y are reversed to find the reciprocal function, the following pattern always holds: The derivative of Derivative Formula for Inverses: evaluated at is equal to the reciprocal of the derivative of evaluated at .

A typical problem using this formula might look like this: Given: Find: Derivative Formula for Inverses:

We can use implicit differentiation to find:

We can use implicit differentiation to find: But so is positive.

We could use the same technique to find and . 1 sec d x dx -

Your calculator contains all six inverse trig functions. However it is occasionally still useful to know the following:

Example Differentiate y = sin-1x2

Example A particle moves a long the x-axis so that its position at any time t>0 is x(t)=tan-1 what is the velocity of the particle when t = 16?

Example Differentiate y = sec-1 (5x4)

Homework Page 162-163/1-17 odd, 19-21, 23-26 p