1. § 4.3
Differentiation of Exponential Functions

2. Section Outline Chain Rule for eg(x)
Working With Differential Equations
Solving Differential Equations at Initial Values
Functions of the form ekx

3. Chain Rule for eg(x)

4. Chain Rule for eg(x) Differentiate. This is the given function.
EXAMPLE
Differentiate.
SOLUTION
This is the given function.
Use the chain rule.
Remove parentheses.
Use the chain rule for exponential functions.

5. Working With Differential Equations
Generally speaking, a differential equation is an equation that contains a derivative.

6. Solving Differential Equations
EXAMPLE
Determine all solutions of the differential equation
SOLUTION
The equation has the form y΄ = ky with k = 1/3. Therefore, any
solution of the equation has the form
where C is a constant.

7. Solving Differential Equations at Initial Values
EXAMPLE
Determine all functions y = f (x) such that y΄ = 3y and f (0) = ½.
SOLUTION
The equation has the form y΄ = ky with k = 3. Therefore,
for some constant C. We also require that f (0) = ½. That is,
So C = ½ and

8. Functions of the form ekx