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

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

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

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

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

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

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

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