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§ 4.3

§ 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

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  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) 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

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