1. § 4.2
The Exponential Function e x

2. Section Outline e
The Derivatives of 2x, bx, and ex

3. The Number e Definition Example
e: An irrational number, approximately equal to , such that the function f (x) = bx has a slope of 1, at x = 0, when b = e

4. The Derivative of 2x

5. Solving Exponential Equations
EXAMPLE
Calculate.
SOLUTION

6. The Derivatives of bx and ex

7. Solving Exponential Equations
EXAMPLE
Find the equation of the tangent line to the curve at (0, 1).
SOLUTION
We must first find the derivative function and then find the value of the derivative at (0, 1). Then we can use the point-slope form of a line to find the desired tangent line equation.
This is the given function.
Differentiate.
Use the quotient rule.

8. Solving Exponential Equations
CONTINUED
Simplify.
Factor.
Simplify the numerator.
Now we evaluate the derivative at x = 0.

9. Solving Exponential Equations
CONTINUED
Now we know a point on the tangent line, (0, 1), and the slope of that line, -1. We will now use the point-slope form of a line to determine the equation of the desired tangent line.
This is the point-slope form of a line.
(x1, y1) = (0, 1) and m = -1.
Simplify.