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# Calc 2.2a

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### Transcript

• 1. 2.2 BASIC DIFFERENTIATION RULES
• To Find:
• Derivative using Constant Rule
• Derivative using Power Rule
• Derivative using Constant Multiple Rule
• Derivative using Sum and Difference Rules
• Derivative for Sine and Cosine
• 2. Makes sense, right?
• 3. Let’s see if we can come up with the power rule.
• f(x) = x
• f(x) = x 2
• c. f(x) = x 3
Do you recognize a pattern?
• 4.
• f(x) = x 5
• b.
• c.
• Sometimes you need to rewrite to different form!
Find the following derivatives:
• 5.
• Ex. 3 p. 109 Find the slope of a graph.
• Find the slope of f(x) = x 4 when
• x = -1
• x = 0
• x = 1
• 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. Informally, this states that constants can be factored out of the differentiation process. Ex 5 p. 110 Using the Constant Multiple Rule
• 8. Ex 5 continued
• 9. Ex 6 p. 111 Using Parentheses when differentiating Original function Rewrite Differentiate Simplify
• 10. This can be expanded to any number of functions Ex 7, p. 111 Using Sum and Difference Rules a. b.
• 11. Proof of derivative of sine:
• 12.
• 13. Last but not least, Ex 8, p112, Derivatives of sine and cosine Function Derivative
• 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)