1. Calculating the Derivative
Chapter 12

2. Ch. 12 Calculating the Derivative
12.1 Techniques for Finding Derivatives
12.2 Derivatives of Products and Quotients
12.3 The Chain Rule
12.4 Derivatives of Exponential Functions
12.5 Derivatives of Logarithmic Functions

3. 12.1 Techniques for Finding the Derivative
Using the definition (below) to calculate the derivative can be very involved.
Section 11.5 introduces rules to simplify the calculation of derivatives.
The simplified process does not change the interpretation of the derivative.

4. Techniques for Finding the Derivative
NOTATIONS FOR THE DERIVATIVE
The derivative of y = f(x) may be written in any of the following ways:
Each of the above represents the “derivative of the function f(x) (or y) with respect to x.”

5. CONSTANT RULE If f(x) = k, where k is any real number, then f ‘(x) = 0
(The derivative of a constant is 0)
If f(x) = 12, then f ‘(x) = 0
If p(t) = , then Dt[p(t)] = 0
If y = 23, then dy/dx = 0

6. POWER RULE If f(x) = xn for any real number n, then f ‘(x) = nxn – 1
(The derivative of f(x) = xn is found by multiplying by the exponent n and decreasing the exponent on x by 1.)
If y = x3, then Dxy = 3x3 – 1 = 3x2
If y = x, then dy/dx = 1x1 – 1 = x0 = 1

7. CONSTANT TIMES A FUNCTION
Let k be a real number. If f ‘(x) exists, then
Dx[kf(x)] = kf ‘(x)
(The derivative of a constant times a function is the constant times the derivative of the function.)

8. A FUNCTION DIVIDED BY A CONSTANT
Let k be a real number. If f ‘(x) exists, then
(The derivative of a function divided by a constant is the derivative of the function divided by the constant.)

9. SUM OR DIFFERENCE RULE
If f(x) = u(x) ± v(x) , and if u’(x) and v’(x) exist, then
f ‘(x) = u’(x) ± v’(x)
(The derivative of a sum or difference of functions is the sum or difference of the derivatives.)
y = 2x5 + 6x3
let u(x) = 2x5 and v(x) = 6x3, then y = u(x) + v(x)
Since u’(x) = 10x4 and v’(x) = 18x2,
dy/dx = 10x4 + 18x2
f(x) = (4x2 – 3x)2
f(x) = 16x4 – 24x3 + 9x2 [(a – b)2 = a2 – 2ab + b2]
f ‘x = 64x3 – 72x2 + 18x

10. Marginal Analysis
Marginal Revenue (MR) is the derivative of the Total Revenue function.
When MR = 0, revenue is maximized!
Marginal Cost (MC) is the derivative of the Total Cost function.
When MC = 0, cost is minimized!
When MR = MC, profit is maximized!
Marginal Product (MP) is the derivative of the Total Product function.
Marginal Propensity to Consume (MPC) is the derivative of the consumption function.

11. Marginal Cost – example
A firm’s total cost in hundreds of dollars to produce x thousand widgets is given by
C(x) = 0.1x2 + 3
Find the marginal cost of producing at x = 10,000 widgets.
MC = C’(x)
C’(x) = 0.2x
Therefore, the marginal cost at x = 10,000 widgets is
MC = 0.2(10) = 2 or $200
After 10,000 widgets have been produced, the cost to produce 1,000 more widgets will be approximately $200

12. Marginal Cost – another example
Average Cost (AC) = TC/Output
If a manufacturer’s average cost equation is
AC = q q (5000 / q, where q = output
Find the marginal cost function
Calculate MC when q = 50

13. Since AC = TC/q, then TC = q(AC)
Differentiating TC gives MC
MC (at q = 50) = (50)2 – 0.04(50) + 5 = $3.75

14. Now You Try Assume that a demand equation is given by
Find the marginal revenue at an output of 1000 units. Hint: Solve the demand equation for p and use R(x) = xp

15. 12.2 Derivatives of Products and Quotients
PRODUCT RULE
If f(x) = u(x) • v(x), and u’(x) and v’(x) both exist, then
f ’(x) = u(x) • v’(x) + v(x) • u’(x)
(The derivative of a product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first.)
Consider f as the product of two functions:
(x2 + 3x) = h(x) and (4x + 5) = g(x),
so, f(x) = h(x)· g(x).

16. f ’(x) = h(x)g’(x) + g(x)h’(x)
PRODUCT RULE
apply the product rule,
f ’(x) = h(x)g’(x) + g(x)h’(x)
f ’(x) = (x2 + 3x) (4) + (4x + 5) (2x + 3)
f ’(x) = 4x2 + 12x +8x2 + 12x + 10x + 15
= 12x2 + 34x + 15
f(x) = h(x)· g(x).

17. Derivatives of Products and Quotients
QUOTIENT RULE
If f(x) = u(x)/v(x), and all indicated derivatives exist, and v(x)  0, then
(The derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.)

18. QUOTIENT RULE
Example: Find f ’(x) if:

19. Quotient Rule and Marginal Revenue
Suppose that a firm’s total revenue function is
R = 1000q / (q + 5).
Find the marginal revenue function.
How much additional revenue will the 46th unit sold bring in? (Calculate MR at q = 45)

20. Now You Try.
A company that manufactures bicycles has determined that a new employee can assemble M(d) bicycles per day after d days of on-the-job training, where
Find the rate of change function for the number of bicycles assembled with respect to time.
Find and interpret M’(2) and M’(5).

21. 12.3 The Chain Rule Composite Functions S = f(I) I = g(t) S = f [g(t)]
The amount of money Sharon saves (S) depends upon her income (I). We can express this as
S = f(I)
Sharon’s income increases each year (t) she continues to work. So,
I = g(t)
The composition of functions f and g:
S = f [g(t)]

22. Composite Functions - example

23. The Chain Rule Used to find the derivative of a composite function.
If y is a function of u, or y = f(u), and if u is a function of x, or u = g(x), then y = f(u) = f[g(x)], and

24. The Chain Rule – example
Think of y = (x2 + 1)3 as a composite of two functions:
y = u3 and u = x2 + 1
Then,
Replace u with x2 + 1

25. Now You Try
Assume that the total revenue from the sale of x television sets is given by
Find the marginal revenue when the following numbers of sets are sold
400
500
600

26. 12.4 Derivatives of Exponential Functions
This section examines four exponential functions:
Derivative of ex
Derivative of ax
Derivative of ag(x)
Derivative of eg(x)

27. Derivatives of Exponential Functions
DERIVATIVE OF ex
Dx[ex] = ex
DERIVATIVE OF ax
Dx[ax] = (ln a)ax
y = 5t

28. Derivatives of Exponential Functions
DERIVATIVE OF ag(x)
Dx[ag(x)] = (ln a) ag(x)g’(x)
Let g(x) = 5x, with g’(x) = 5. Then,

29. Derivatives of Exponential Functions
DERIVATIVE OF eg(x)
Dx[eg(x)] = eg(x)g’(x)
Let g(x) = 2x, with g’(x) = 2. Then,

30. Example
The sales of new personal computers (in thousands) are given by
S(t) = 100 – 90e- 0.3t
where t represents time in years.
Find the rate of change of sales
After 1 year
After 5 years

31. Example 27e -0.3(1) = 20 (or 20,000) 27e -0.3(5) = 6 (or 6,000)
Let g(x) = -0.3t, with g’(x) = Then,
27e -0.3(1) = 20 (or 20,000)
27e -0.3(5) = 6 (or 6,000)

32. 12.5 Derivatives of Logarithmic Functions
DERIVATIVE OF ln |x|

33. Derivatives of Logarithmic Functions
DERIVATIVE OF [ln |g(x)|]

34. Now You Try
Suppose the demand function for x thousand units of a certain item is
where p is in dollars.
Find the marginal revenue if quantity demanded is 8000 units.

35. 
Chapter 12
End