Section 12.1 Techniques for Finding Derivative

Constant Rule

Power Rule

Sum and Difference Rule

0d

cdx

1n ndx n x

dx

( ) ( ) '( ) '( )d

f x g x f x g xdx

Examples

Find y’ for the function.

42

15y x xx

4 2y x x x

Constant Times a Function

Examples:

( ) '( )d

c f x c f xdx

Find dy/dx for the function.

34

5 42 3y x x

x x

5 3 1.5 2 710 3 55

7

xy x x

xx

Marginal Functions

Example: The total cost in dollars incurred each week for manufacturing q refrigerators is given by the total cost function:

C(q) =8000 + 200q – 0.25q2

What is the actual cost incurred for manufacturing the 251st refrigerator?

Marginal Cost

Marginal cost is the cost incurred in producing an additional unit of a certain item given that the plant is already at a certain level of operation.

Mathematically, marginal cost is the rate of change of the total cost function with respect to x

derivative of the cost function. If C(x) is a total cost function, then the derivative C’(x) is called the marginal cost function.

Example

The total cost in dollars incurred each week for manufacturing q refrigerators is given by the total cost function:

C(q) =8000 + 200q – 0.25q2

Find the marginal cost for producing 251 refrigerators.

Marginal Revenue / Profit

If R(x) is a revenue function, then the derivative R’(x) is called the marginal revenue function.

If P(x) is a profit function, then the derivative P’(x) is called the marginal profit function.

Recall R(q) = pq or R(x) = px and P = R - C

Example

The total cost C(q) =8000 + 200q – 0.25q2

Suppose the demand equation for the refrigerators each week is given by

q = 9000 – 5p.

Find the marginal revenue for the production level of

a) 200 units.

b) 400 units.