1. Warmup 4/7/16 The Bible tells us that someday Jesus is going to return. If that’s true, then why do you think He’s taking so long? To determine the derivative of inverse functions pp 473: 4, 5, 6, 7 Objective Tonight’s Homework

2. Homework Help Let’s spend the first 10 minutes of class going over any problems with which you need help.

3. Notes on Derivatives of Inverse Functions Let’s imagine a simple function. y = x 2 Let’s take the slope of this function at (2, 4). It would be a slope of 4.

4. Notes on Derivatives of Inverse Functions Let’s imagine a simple function. y = x 2 Let’s take the slope of this function at (2, 4). It would be a slope of 4. Now let’s look at the inverse function. x = y 2 We can rearrange this as y = +√ x If we take the slope at (4, 2), we get ¼. So what does this tell us?

5. Notes on Derivatives of Inverse Functions It says the following: The slope of the function at (a, b) equals the reciprocal of the slope of the inverse function at (b, a). Or written mathematically: (f -1 )’(x) = 1 _ f ’(f -1 (x))

6. Notes on Derivatives of Inverse Functions It says the following: The slope of the function at (a, b) equals the reciprocal of the slope of the inverse function at (b, a). Or written mathematically: (f -1 )’(x) = If we want to know the slope of the inverse function at a point, all we have to do is plug the inverse function into the derivative of the function at that point and take the reciprocal. 1 _ f ’(f -1 (x))

7. Notes on Derivatives of Inverse Functions Example: Look at the function f(x) = x 3 + x – 1. Find the slope of the inverse function f -1 at the point (-1, 0).

8. Notes on Derivatives of Inverse Functions Example: Look at the function f(x) = x 3 + x – 1. Find the slope of the inverse function f -1 at the point (-1, 0). To get the slope of the inverse function at this point, we need the inverse and derivative of our original function. y = x 3 + x – 1Original Function x = y 3 + y – 1Inverse Function y = x 3 + x – 1Original Function y’ = 3x 2 + 1Derivative

9. Notes on Derivatives of Inverse Functions We can now plug these into our formula to get the slope of the inverse at the point (-1, 0). (f -1 )’(-1) = (f -1 )’(-1) = 1/1 (f -1 )’(-1) = 1 1 _ f ’(f -1 (-1)) 1 _ f ’(-1 = y 3 + y – 1) 1 _ f ’(0) 1 _ y’ = 3(0) 2 + 1

10. Group Practice Look at the example problems on pages 471 through 473. Make sure the examples make sense. Work through them with a friend. Then look at the homework tonight and see if there are any problems you think will be hard. Now is the time to ask a friend or the teacher for help! pp 473: 4, 5, 6, 7

11. Exit Question What's the slope of the inverse of y = x 2 at x=1? a) 0 b) 1/2 c) infinite d) undefined e) Not enough information f) None of the above