Derivatives of Inverse Functions Lewis and Clark Caverns, Montana Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College 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:

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