1. Chapter 12
Additional Differentiation Topics

2. 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.

3. 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)

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

5. 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:

6. 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 .

7. 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:

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

9. 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 .

10. 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

11. 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. 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

13. 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.

14. 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:

15. 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:

16. 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.

17. 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:

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

19. 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

20. 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

21. 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,

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

23. 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).

24. 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:

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