1. Differentiation 3 Basic Rules of Differentiation The Product and Quotient Rules The Chain Rule Marginal Functions in Economics Higher-Order Derivatives

2. Basic Differentiation Rules 1. Ex. 2. Ex.

3. Basic Differentiation Rules 3. Ex. 4. Ex.

4. More Differentiation Rules 5. Ex. Product Rule Derivative of the first function Derivative of the second function

5. More Differentiation Rules 6.Quotient Rule Sometimes remembered as:

6. More Differentiation Rules 6. Ex. Quotient Rule (cont.) Derivative of the numerator Derivative of the denominator

7. More Differentiation Rules 7. The Chain Rule Note: h(x) is a composite function. Another Version:

8. More Differentiation Rules The General Power Rule: Ex. The Chain Rule leads to

9. Chain Rule Example Ex.

10. Chain Rule Example Ex. Sub in for u

11. Marginal Functions The Marginal Cost Function approximates the change in the actual cost of producing an additional unit. The Marginal Average Cost Function measures the rate of change of the average cost function with respect to the number of units produced. The Marginal Profit Function measures the rate of change of the profit function. It approximates the profit from the sale of an additional unit. The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit.

12. Example The monthly demand for T-shirts is given by where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The monthly cost function for these T-shirts is 1. Find the revenue and profit functions. 2. Find the marginal cost, marginal revenue, and marginal profit functions.

13. Solution 1. Find the revenue and profit functions. 2. Find the marginal cost, marginal revenue, and marginal profit functions. Revenue = xp Profit = revenue – cost Marginal Cost =

14. Solution 2. (cont.) Find the marginal revenue and marginal profit functions. Marginal revenue = Marginal profit =

15. Higher Derivatives The second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative. DerivativeNotations Second Third Fourth nth

16. Example of Higher Derivatives Givenfind

17. Example of Higher Derivatives Givenfind