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Chapter 12 Additional Differentiation Topics

Chapter 12 Additional Differentiation Topics. Chapter 12: Additional Differentiation Topics Chapter Objectives. To develop a differentiation formula for y = ln u . To develop a differentiation formula for y = e u . To give a mathematical analysis of the economic concept of elasticity.

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Chapter 12 Additional Differentiation Topics

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  1. Chapter 12 Additional Differentiation Topics

  2. Chapter 12: Additional Differentiation Topics • Chapter Objectives • To develop a differentiation formula for y = ln u. • To develop a differentiation formula for y = eu. • To give a mathematical analysis of the economic concept of elasticity. • To discuss the notion of a function defined implicitly. • To show how to differentiate a function of the form uv. • To approximate real roots of an equation by using calculus. • To find higher-order derivatives both directly and implicitly.

  3. Chapter 12: Additional Differentiation Topics • Chapter Outline 12.1) Derivatives of Logarithmic Functions Derivatives of Exponential Functions Elasticity of Demand Implicit Differentiation Logarithmic Differentiation Newton’s Method Higher-Order Derivatives 12.2) 12.3) 12.4) 12.5) 12.6) 12.7)

  4. Chapter 12: Additional Differentiation Topics • 12.1 Derivatives of Logarithmic Functions • The derivatives of log functions are:

  5. Chapter 12: Additional Differentiation Topics • 12.1 Derivatives of Logarithmic Functions • Example 1 – Differentiating Functions Involving ln x a.Differentiate f(x) = 5 ln x. Solution: b.Differentiate . Solution:

  6. Chapter 12: Additional Differentiation Topics • 12.1 Derivatives of Logarithmic Functions • Example 3 – Rewriting Logarithmic Functions before Differentiating a. Find dy/dx if . Solution: b. Find f’(p) if . Solution:

  7. Chapter 12: Additional Differentiation Topics • 12.1 Derivatives of Logarithmic Functions • Example 5 – Differentiating a Logarithmic Function to the Base 2 Procedure to Differentiate logbu • Convert logbu to and then differentiate. Differentiate y = log2x. Solution:

  8. Chapter 12: Additional Differentiation Topics • 12.2 Derivatives of Exponential Functions • The derivatives of exponential functions are:

  9. Chapter 12: Additional Differentiation Topics • 12.2 Derivatives of Exponential Functions • Example 1 – Differentiating Functions Involving ex • Find . • Solution: • b.If y = , find . • Solution: • c.Find y’ when . • Solution:

  10. Chapter 12: Additional Differentiation Topics • 12.2 Derivatives of Exponential Functions • Example 3 – The Normal-Distribution Density Function Determine the rate of change of y with respect to x when x = μ + σ. Solution: The rate of change is

  11. Chapter 12: Additional Differentiation Topics • 12.2 Derivatives of Exponential Functions • Example 5 – Differentiating Different Forms • Example 6 – Differentiating Power Functions Again Find . Solution: Prove d/dx(xa) = axa−1. Solution:

  12. Chapter 12: Additional Differentiation Topics • 12.3 Elasticity of Demand • Example 1 – Finding Point Elasticity of Demand • Point elasticity of demandη is where p is price and q is quantity. Determine the point elasticity of the demand equation Solution: We have

  13. Chapter 12: Additional Differentiation Topics • 12.4 Implicit Differentiation Implicit Differentiation Procedure • Differentiate both sides. • Collect all dy/dx terms on one side and other terms on the other side. • Factor dy/dx terms. • Solve for dy/dx.

  14. Chapter 12: Additional Differentiation Topics • 12.4 Implicit Differentiation • Example 1 – Implicit Differentiation Find dy/dx by implicit differentiation if . Solution:

  15. Chapter 12: Additional Differentiation Topics • 12.4 Implicit Differentiation • Example 3 – Implicit Differentiation Find the slope of the curve at (1,2). Solution:

  16. Chapter 12: Additional Differentiation Topics • 12.5 Logarithmic Differentiation Logarithmic Differentiation Procedure • Take the natural logarithm of both sides which gives . • Simplify In (f(x))by using properties of logarithms. • Differentiate both sides with respect to x. • Solve for dy/dx. • Express the answer in terms of x only.

  17. Chapter 12: Additional Differentiation Topics • 12.5 Logarithmic Differentiation • Example 1 – Logarithmic Differentiation Find y’if . Solution:

  18. Chapter 12: Additional Differentiation Topics • 12.5 Logarithmic Differentiation • Example 1 – Logarithmic Differentiation Solution (continued):

  19. Chapter 12: Additional Differentiation Topics • 12.5 Logarithmic Differentiation • Example 3 – Relative Rate of Change of a Product Show that the relative rate of change of a product is the sum of the relative rates of change of its factors. Use this result to express the percentage rate of change in revenue in terms of the percentage rate of change in price. Solution: Rate of change of a function r is

  20. Chapter 12: Additional Differentiation Topics • 12.6 Newton’s Method • Example 1 – Approximating a Root by Newton’s Method Newton’s method: Approximate the root of x4 − 4x + 1 = 0 that lies between 0 and 1. Continue the approximation procedure until two successive approximations differ by less than 0.0001.

  21. Chapter 12: Additional Differentiation Topics • 12.6 Newton’s Method • Example 1 – Approximating a Root by Newton’s Method Solution: Letting , we have Since f (0) is closer to 0, we choose 0 to be our first x1. Thus,

  22. Chapter 12: Additional Differentiation Topics • 12.7 Higher-Order Derivatives For higher-order derivatives:

  23. Chapter 12: Additional Differentiation Topics • 12.7 Higher-Order Derivatives • Example 1 – Finding Higher-Order Derivatives a. If , find all higher-order derivatives. Solution: b. If f(x) = 7, find f(x). Solution:

  24. Chapter 12: Additional Differentiation Topics • 12.7 Higher-Order Derivatives • Example 3 – Evaluating a Second-Order Derivative • Example 5 – Higher-Order Implicit Differentiation Solution: Solution:

  25. Chapter 12: Additional Differentiation Topics • 12.7 Higher-Order Derivatives • Example 5 – Higher-Order Implicit Differentiation Solution (continued):

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