1. Chapter 7 Rules of Differentiation
Isaac Newton and Gottfried Leibniz

2. 7.1 Review: Derivative and Derivative Rules
Review: Definition of derivative.
Applying this definition, we review the 9 rules of differentiation. First, the constant rule:
The derivative of a constant function is zero for all values of x.

3. 7.1 A Review of the 9 Rules of Differentiation

4. 7.1 A Review of the 9 Rules of Differentiation4 4

5. 7.1.1 Power-Function Rule Example: Let Total Revenue (R) be:
R = 15 Q – Q2 => dR/dQ = MR =15 – 2Q.
As Q increases R increases (as long as Q>7.5).

6. 7.1.2 Exponential-Function Rule6 6

7. 7.1.2 Exponential-Function Rule: Joke
A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying "I differentiate you!"
One day he met a new patient; and true to form he stared at him and said "I differentiate you!", but for once, his victim's expression didn't change.
Surprised, the mathematician collected all his energy, stared fiercely at the new patient and said loudly "I differentiate you!", but still the other man had no reaction. Finally, in frustration, the mathematician screamed out "I DIFFERENTIATE YOU!"
The new patient calmly looked up and said, "You can differentiate me all you like: I'm ex."

8. 7.2.1 Sum or difference rule

9. 7.2.2 Product rule
The derivative of the product of two functions is equal to the second function times the derivative of the first plus the first function times the derivative of the second.
Example: Marginal Revenue

10. 7.2.4 Quotient rule

11. 7.3.1 Chain rule

12. 7.3.1 Chain rule

13. 7.3.1 Chain rule: Application - Log rule
Consider h(x) = eln(x) = x
Then, h’(x) = 1.
Let’s apply Chain rule to h(x) 13 13

14. 7.3.1 Chain rule and its relation to total differential

15. 7.3.2 Inverse function rule
Let y=f(x) be a differentiable strictly monotonic function.
Note: A monotonic function is one in which a given value of x yields a unique value of y, and given a value of y will yield a unique value of x (a one-to-one mapping). These types of functions have a defined inverse.

16. 7.3.2 Inverse-function rule
This property of one to one mapping is unique to the class of functions known as monotonic functions:
Recall the definition of a function function: one y for each x monotonic function: one x for each y (inverse function)
if x1 > x2  f(x1) > f(x2) monotonically increasing
Qs = b0 + b1P supply function (where b1 > 0)
P = -b0/b1 + (1/b1)Qs inverse supply function
if x1 > x2  f(x1) < f(x2) monotonically decreasing
Qd = a0 - a1P demand function (where a1 > 0)
P = a0/a1 - (1/a1)Qd inverse demand function

17. 7.4 Multivariate Calculus: Partial Differentiation
Now, y depends on several variables: x1, x2, …, xn.
The derivative of y w.r.t. a one of the variables –and the other variables are held constant- is called a partial derivative.

18. 7.4.1 Partial derivatives: Example

19. 7.4.2 Partial differentiation: Market Model

20. 7.4.2 Partial differentiation: Market Model
Using linear algebra, we have:

21. 7.6 Note on Jacobian Determinants
The Jacobian is the matrix of first partial derivatives.
Use Jacobian determinants to test the existence of functional dependence between the functions |J|
Not limited to linear functions as |A| (special case of |J|
If |J| = 0, then the non-linear or linear functions are dependent and a solution does not exist.

22. 7.7 Second and Higher Derivatives
Derivative of a derivative
Given y = f(x)
The first derivative f '(x) or dy/dx is itself a function of x, it should be differentiable with respect to x, provided that it is continuous and smooth.
The result of this differentiation is known as the second derivative of the function f and is denoted as f ''(x) or d2y/dx2.
The second derivative can be differentiated with respect to x again to produce a third derivative:
f '''(x) and so on to f(n)(x) or dny/dxn
This process can be continued to produce an n-th derivative. 22 22

23. 7.7 Example: 1st. 2nd and 3rd derivatives
Graphically: 23 23

24. 7.7 Example: Higher Order Derivatives24 24

25. 7.7 Interpretation of the second derivative
f '(x) measures the rate of change of a function
e.g., whether the slope is increasing or decreasing
f ''(x) measures the rate of change in the rate of change of a function
e.g., whether the slope is increasing or decreasing at an increasing or decreasing rate
how the curve tends to bend itself
Utility functions are increasing in consumption f '(x)>0. But they differ by the rate of change in f '(x)>0; that is, they differ on f ''(x).
f ''(x) >0, increasing f '(x)>0
f ''(x) =0, constant f '(x)>0
f ''(x) <0, decreasing f '(x)>0 (usual assumption) 25 25

26. 7.7 Strict concavity and convexity
Strictly concave: if we pick any pair of points M and N on its curve and joint them by a straight line, the line segment MN must lie entirely below the curve, except at points MN.
A strictly concave curve can never contain a linear segment anywhere (if it does it’s just concave, not strictly concave).
Test: if f "(x) is negative for all x, then strictly concave.
Strictly convexity: if we pick any pair of points M and N on its curve and joint them by a straight line, the line segment MN must lie entirely above the curve, except at points MN.
A strictly convex curve can never contain a linear segment anywhere (if it does it’s just convex, not strictly convex).
Test: if f "(x) is positive for all x, then strictly convex. 26 26

27. Figure 7.6 Concave and Convex Functions

28. 7.7 Concavity and Convexity:  & 
If f "(x) < 0 for all x, then strictly concave.
If f "(x) > 0 for all x, then strictly convex.  
Application: Risk Aversion
AP = Arrow-Pratt measure = -U’’(x)/U’(x)
Let U(x) = β ln(x) (β >0)
U’(x) = β/x > 0;
U’’(x) = -β /x-2 < 0
AP = 1/x =>as x (wealth) increases, risk aversion decreases. 28 28

29. Figure 7.5 Logarithmic Utility Function

30. Figure 7.7 Utility Functions for Risk-Averse and Risk-Loving Individuals

31. 7.8 Taylor series of a polynomial functions
The Taylor series is a representation of a (infinitely differentiable) function as an infinite sum of terms calculated from the values of its derivatives at a single point, x0. It may be regarded as the limit of the Taylor polynomials. If the series is centered at zero, x0=0, the series is called a Maclaurin series.
Brook Taylor (1685 – 1731) 31 31

32. 7.8 Taylor series of a polynomial functions32 32

33. 7.8 Taylor series of a polynomial functions
Taylor series work very well for polynomials; the exponential function ex and the sine and cosine functions. (They are all examples of entire functions –i.e., f(x) equals its Taylor series everywhere).
Taylor series do not always work well. For example, for the logarithm function, the Taylor series do not converge if x is far from x0.
Log approximation around 0: 33 33

34. 7.8 Maclaurin Series of a polynomial function34 34

35. 7.8 Maclaurin Series of a Polynomial Function35 35

36. 7.8 Maclaurin Series of ex 36 36

37. 7.8 Geometric series 37 37

38. 7.8 Geometric series 38 38

39. 7.8 Geometric series: Approximating A-139 39

40. 7.8 Application: Geometric series & PV Models
A stock price (P) is equal to the discounted some of all futures dividends. Assume dividends are constant (d) and the discount rate is r. Then: 40 40

41. 7.8 Taylor Series of a Polynomial Function41 41

42. 7.8 Taylor Series of a Polynomial Function
Do a first-order Taylor series expansion of f(x) around x0=1.
f(x)=5+2x + x2 f’ (x0=1) = 8
f’ (x) = 2 + 2x f ’(x0=1) = 4
Taylor’s series formula:
First-order Taylor series: f(x) = (x-1) + R = 4 + 4x + R
Let’s check the approximation error
f(x)=5+2x + x2 f(x)=4+4x + R
x0=1 f(1) = 8 f(1) = 8
x0=1.1 f(1.1) = f(1.1) = 8.4
x0 =1.2 f(1.1) = f(1.1) = 8.8 42 42

43. 7.8 Taylor Series of a Polynomial Function43 43

44. Q: What is the first derivative of a cow? A: Prime Rib!