Chapter 12 Additional Differentiation Topics

Chapter Objectives To develop a differentiation formula for y = ln u. 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.

Chapter Outline Derivatives of Logarithmic Functions 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)

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

a. Differentiate f(x) = 5 ln x. Solution: 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:

a. Find dy/dx if . Solution: b. Find f’(p) if . 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 .

Procedure to Differentiate logbu 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:

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

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

Determine the rate of change of y with respect to x when x = μ + σ. 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

Find . Solution: Prove d/dx(xa) = axa−1. Solution: 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:

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

12.4 Implicit Differentiation 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.

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

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

12.5 Logarithmic Differentiation 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.

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

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

Solution: Rate of change of a function r is 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

12.6 Newton’s Method Newton’s method: 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.

Solution: Letting , we have 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,

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

a. If , find all higher-order derivatives. Solution: 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: Solution: Example 3 – Evaluating a Second-Order Derivative 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:

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