Download presentation

Presentation is loading. Please wait.

Published byAugusta Heath Modified over 6 years ago

1
7.3* The Natural Exponential Function INVERSE FUNCTIONS In this section, we will learn about: The natural exponential function and its properties.

2
LAWS OF EXPONENTS If x and y are real numbers and r is rational, then 1. e x+y = e x e y 2. e x-y = e x /e y 3. (e x ) r = e rx Laws 7

3
The natural exponential function has the remarkable property that it is its own derivative. DIFFERENTIATION Formula 8

4
The function y = e x is differentiable because it is the inverse function of y = l n x. We know this is differentiable with nonzero derivative. To find its derivative, we use the inverse function method. Formula 8—Proof DIFFERENTIATION

5
Let y = e x. Then, l n y = x and, differentiating this latter equation implicitly with respect to x, we get: Formula 8—Proof DIFFERENTIATION

6
Differentiate the function y = e tan x. To use the Chain Rule, we let u = tan x. Then, we have y = e u. Hence, DIFFERENTIATION Example 4

7
In general, if we combine Formula 8 with the Chain Rule, as in Example 4, we get: DIFFERENTIATION Formula 9

8
Find y’ if y = e -4x sin 5x. Using Formula 9 and the Product Rule, we have: DIFFERENTIATION Example 5

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google