1. Differentiation What is a Function? A function in math is the process of showing a relationship between the input and output of a problem, usually written as an equation. It is designed to show relationships between a set of numbers and a different set of numbers "f(x) =... " is the classic way of writing a function. 1

2. Differentiation Input, Relationship, Output I will show you many ways to think about functions, but there will always be three main parts: –The input –The relationship –The output 2

3. Differentiation x 2 (squaring) is a function x 3 +1 is also a function 3

4. Differentiation InputRelationshipOutput 0× 20 1 2 7 14 10× 220... 4

5. Differentiation Example: with f(x) = x 2 : an input of 4 becomes an output of 16. In fact we can write f(4) = 16. The "x" is Just a Place-Holder! Don't get too concerned about "x", it is just there to show you where the input goes and what happens to it. It could be anything! 5

6. Differentiation So this function: f(x) = 1 - x + x 2 Would be the same function if I wrote: –f(q) = 1 - q + q2 –h(A) = 1 - A + A2 –w(θ) = 1 - θ + θ2 –It is just there so you know where to put the values: f(2) = 1 - 2 + 22 = 3 6

7. 7 Rules of Differentiation for a Function of One Variable Constant-Function Rule Power-Function Rule Power-Function Rule Generalized

8. 8 Rules of Differentiation and Their Use in Comparative Statics Rules of Differentiation for a Function of One Variable Rules of Differentiation Involving Two or More Functions of the Same Variable Rules of Differentiation Involving Functions of Different Variables Partial Differentiation

9. 9 Rules of Differentiation for a Function of One Variable

10. 10 Constant-Function Rule

11. 11 Power-Function Rule

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14. 14 Power-Function Rule

15. 15 Power-Function Rule

16. 16 Rules of Differentiation Involving Two or More Functions of the Same Variable Sum-difference rule Product rule Finding marginal-revenue function from average-revenue function Quotient rule Relationship between marginal-cost and average-cost functions

17. 17 Sum or difference rule

18. 18 Product rule

19. 19 Finding marginal-revenue function from average- revenue function using the product rule

20. 20 Relationship between marginal-cost and average- cost functions C MC AC Q

21. 21 Chain rule

22. 22 Chain rule

23. 23 Partial derivatives