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Differentiation of Exponential Functions and Solving Differential Equations

This section covers the differentiation of exponential functions, emphasizing the chain rule application for functions of the form ( e^{g(x)} ). We also delve into differential equations, defining them as equations containing derivatives. Specific examples illustrate how to solve differential equations from initial values, highlighting methods to find all solutions for equations like ( y' = ky ) and ( y' = 3y ) with given initial conditions. Understanding these concepts is crucial for analyzing dynamic systems in mathematics and applied sciences.

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Differentiation of Exponential Functions and Solving Differential Equations

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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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