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# Calc 2.2a

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### Calc 2.2a

1. 1. 2.2 BASIC DIFFERENTIATION RULES <ul><li>To Find: </li></ul><ul><li>Derivative using Constant Rule </li></ul><ul><li>Derivative using Power Rule </li></ul><ul><li>Derivative using Constant Multiple Rule </li></ul><ul><li>Derivative using Sum and Difference Rules </li></ul><ul><li>Derivative for Sine and Cosine </li></ul>
2. 2. Makes sense, right?
3. 3. Let’s see if we can come up with the power rule. <ul><li>f(x) = x </li></ul><ul><li>f(x) = x 2 </li></ul><ul><li>c. f(x) = x 3 </li></ul>Do you recognize a pattern?
4. 4. <ul><li>f(x) = x 5 </li></ul><ul><li>b. </li></ul><ul><li>c. </li></ul><ul><li>Sometimes you need to rewrite to different form! </li></ul>Find the following derivatives:
5. 5. <ul><li>Ex. 3 p. 109 Find the slope of a graph. </li></ul><ul><li>Find the slope of f(x) = x 4 when </li></ul><ul><li>x = -1 </li></ul><ul><li>x = 0 </li></ul><ul><li>x = 1 </li></ul>
6. 6. Ex 4 p. 109 Finding an Equation of a Tangent Line Find the equation of the tangent line to the graph of f(x) = x 3 when x = -2 To find an equation of a line, we need a point and a slope. The point we are looking at is (-2, f(-2)). In other words, find the y-value in the original function! f(-2) = (-2) 3 = -8. So our point of tangency is (-2, -8) Next we need a slope. Find the derivative & evaluate. f ‘(x) = 3x 2 so find f ‘(-2) = 3 ٠ (-2) 2 =12 Equation: (y – (-8)) = 12(x –(-2)) So y = 12x +16 is the equation of the tangent line.
7. 7. Informally, this states that constants can be factored out of the differentiation process. Ex 5 p. 110 Using the Constant Multiple Rule
8. 8. Ex 5 continued
9. 9. Ex 6 p. 111 Using Parentheses when differentiating Original function Rewrite Differentiate Simplify
10. 10. This can be expanded to any number of functions Ex 7, p. 111 Using Sum and Difference Rules a. b.
11. 11. Proof of derivative of sine:
12. 13. Last but not least, Ex 8, p112, Derivatives of sine and cosine Function Derivative
13. 14. Assign: 2.2a p. 115 #1-65 every other odd Heads up – each of you will need to create a derivative project – something that you will use to remember all the derivative rules we learn in this chapter. This will be due Monday Oct 17. See paper for details. (online too)