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# 125 2.2 and 2.3

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### 125 2.2 and 2.3

1. 1. 2.2 and some of 2.3 Some Differentiation Formulas <ul><li>Derivative of a Constant </li></ul><ul><li>Power Rule </li></ul><ul><li>Evaluation of a Derivative </li></ul><ul><li>Leibniz Notation </li></ul><ul><li>Derivatives in Business and Economics </li></ul>
2. 2. A. Derivative of a Constant <ul><li>The derivative of a constant is _______. </li></ul><ul><li>Examples: </li></ul>
3. 3. B. Power Rule <ul><li>The power comes down front with the coefficient, and you deduct one from the power. Here’s the rule: </li></ul>
4. 4. Power Rule <ul><li>The power comes down front with the coefficient, and you deduct one from the power. Here are examples: </li></ul>
5. 5. What’s the derivative of x ? <ul><li>What about x -5 ? </li></ul>
6. 6. What’s the derivative of x 1/4 ? <ul><li>What’s the derivative of 8 x -1/2 ? </li></ul>
7. 7. What’s the derivative of 7x ? <ul><li>What’s the derivative of x 3 + x 5 ? </li></ul>
8. 8. What’s the derivative of x 3 - x 5 ? <ul><li>What’s the derivative of 5 x -2 - 6x 1/3 + 4? </li></ul>
9. 9. What’s the derivative of 4 x -3 - 3x 1/4 + 171? <ul><li>What’s the derivative of -4 x 1/2 - 6x -3 + x? </li></ul>
10. 10. C. Evaluation of a Derivative <ul><li>Evaluate the derivative of f(x)= x 2 for x = 3. </li></ul><ul><li>Evaluate the derivative of f(x) = x for x = 3. </li></ul>
11. 11.
12. 12.
13. 13.
14. 14. Remember when your algebra teacher taught you this: <ul><li>Rewrite this so that it has no denominators: </li></ul>
15. 15. The reason we need to that is … <ul><li>We don’t yet know how to differentiate </li></ul><ul><li>But we know to differentiate this: </li></ul><ul><li>So rewriting it will be your very first step! </li></ul>
16. 16. You try differentiating:
17. 17. DON’T FORGET WHAT A DERIVATIVE IS <ul><li>It’s a function for the slope of the tangent line. </li></ul><ul><li>If you plug in a value for x. Let’s say, you find the derivative of f and call it f prime. Then suppose you plug in x = 3 into f prime and get 4. So that f’(3) = 4. What does this mean? </li></ul><ul><li>THE LINE THAT IS TANGENT TO f AT THE POINT x=3 HAS A SLOPE OF 4. </li></ul>
18. 18. D. Leibniz Notation
19. 19. E. Derivatives in Business & Economics <ul><li>COST FUNCTIONS: </li></ul><ul><li>C(x) is a function for </li></ul><ul><li>“ total cost of producing x units.” </li></ul><ul><li>MC(x) is for “MARGINAL COST” and it is the same as C’(x), the derivative of C(x). </li></ul><ul><li>Marginal Cost gives you COST PER UNIT. </li></ul>
20. 20. REVENUE FUNCTIONS: <ul><li>R(x) is a function for the total revenue from selling x units. </li></ul><ul><li>MR(x) is for “MARGINAL REVENUE,” and it is the same as R’(x), the derivative of R(x). </li></ul><ul><li>Marginal Revenue gives you REVENUE PER UNIT. </li></ul>
21. 21. PROFIT FUNCTIONS: <ul><li>P(x) is a function for the total profit from producing and selling x units. </li></ul><ul><li>Profit = Revenue minus Cost </li></ul><ul><li>P(x) = R(x) – C(x) </li></ul><ul><li>MP(x) = P’(x) = Marginal Profit = Profit per item </li></ul>