§ 4.3

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# § 4.3 - PowerPoint PPT Presentation

§ 4.3. Differentiation of Exponential Functions. Section Outline. Chain Rule for e g ( x ) Working With Differential Equations Solving Differential Equations at Initial Values Functions of the form e kx. Chain Rule for e g ( x ). Chain Rule for e g ( x ). EXAMPLE. Differentiate.

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§4.3

Differentiation of Exponential Functions

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

Chain Rule for eg(x)

EXAMPLE

Differentiate.

SOLUTION

This is the given function.

Use the chain rule.

Remove parentheses.

Use the chain rule for exponential functions.

Working With Differential Equations

Generally speaking, a differential equation is an equation that contains a derivative.

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.

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