1. 3.8 Derivatives of Inverse Functions

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

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

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

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

6. Let’s try slightly different notation:
In other words, g is the inverse of f
If at the point (a,b)
Then
at the point (b,a)
Slopes are reciprocals.
Onward…

7. We know that
and that
But what about…
Let’s try writing the function like this:
and taking the natural log of both sides…

8. This is called Logarithmic Differentiationp