Section 3.8

Derivatives of Inverse Functions Theorem: If is differentiable at every point of an interval I and is never zero on I, then has an inverse and is differentiable at every point of the interval I.

Derivatives of Inverse Functions y x The slopes of inverse functions are reciprocals, at the corresponding points… in math symbols

Derivatives of Inverse Functions Let. Given that the point is on the graph of, find the slope of the inverse of at. Our new rule: The slope of at is the reciprocal of the slope of at.

First, recall the graph: x y –11 So, should this function be differentiable across its entire domain???  Everywhere except at x = –1 or 1 Derivative of the Arcsine

If is a differentiable function of with, applying the Chain Rule:

Derivative of the Arctangent

If is a differentiable function of, again using the Chain Rule form:

Derivative of the Arcsecant

If is a differentiable function of with, and “chaining” once again, we have:

Derivative of the Others TTTThe derivatives of the inverse cofunctions are the opposites (negatives) of the derivatives of the corresponding inverse functions Inverse Function – Inverse Cofunction Identities:

Guided Practice Find if

Guided Practice Find if

Guided Practice A particle moves along the x-axis so that its position at any time is. What is the velocity of the particle when ? First, find the general equation for velocity:

Guided Practice A particle moves along the x-axis so that its position at any time is. What is the velocity of the particle when ? Now, at the particular time: