3.8 Derivatives of Inverse Trig Functions Lewis and Clark Caverns, Montana Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1993

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:

A function has an inverse only if it is one-to-one. We remember that the graph of a one-to-one function passes the horizontal line test as well as the vertical line test. We notice that if a graph fails the horizontal line test, it must have at least one point on the graph where the slope is zero. one-to-one not one-to-one

Now that we know that we can use the derivative to find the slope of a function, this observation leads to the following theorem: Derivatives of Inverse functions: If f is differentiable at every point of an interval I and df/dx is never zero on I, then f has an inverse and f -1 is differentiable at every point of the interval f(I).

Example: Does have an inverse? Since is never zero, must pass the horizontal line test, so it must have an inverse.

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 -

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