## November 12, 2006

### Scribe Post: Day 41

NOTE that all the derivative rules are in bold and red. All the calculations are in bold. All explanations are in normal text. I will state the rule first, then go through the steps that show how to get it.
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DERIVATIVE RULES
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There are many ways to solve derivatives. Just in our class, we learned about 8-9 ways. In this unit, we are going to find the rules that help us solve for derivatives super easily. We are going to find the derivative rules for Linear Functions (y = mx + b), Constant Functions (y = c), Functions with Constant Variables ( kf(x) ), Exponential Functions (xn), and sum and difference functions (-+).
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Definition of a derivative:
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f'(x) = limh-->0 [f(x+h) - f(x)]/h
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In order to find the derivative rule, we must:
(1) Take the general equation for the function, and
(2) substitute in (x+h) wherever there's an x.
(3) We plug in what we get from step 2 to "f(x+h)" on the definition of a derivative.
(4) We reduce the terms we get as much as possible to get the rule.
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Linear Functions
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f'(x) = m
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f(x) = mx + b
f(x+h) = m(x+h) + b
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f'(x) = [m(x+h) + b] - mx + b
= (mx + mh + b - mx + b)
= mh/h
= m
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Constant Functions
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f'(x) = 0
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f(x) = c
f(x+h) = c
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f'(x) = limh-->0 (c - c)/h
= limh-->0 0
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Functions with Constant Variables
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f'(x) = kf'(x)
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F(x) = kf(x)
F(x+h) = kf(x+h)
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F'(x) = limh-->0 F(x+h) - F(x)/h
= limh-->0 kf(x+h) - kf(x)/h
= limh-->0 k[f(x+h) - f(x)]/h
= limh-->0 klimh-->0 f(x+h) - f(x)/h
= k limh-->0 f(x+h) - f(x)/h
We know that this is the definition for a derivative, so wherever you find this, you can just subsitute f'(x)
= kf'(x)
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This rule is important, because when a function has a constant, we can just mulitply the derivative by the constant.
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This is all for now. I still have to do the rules for polynomial functions (e.g. x3, x4 etc.) and sum and difference functions, but it's soooo hard to do on the computer. Sorry for the inconvencience guys; I tried. There's lots of fractions over fractions, fractions to polynomial exponents and so forth. It just gets really confusing. So I'll just neatly write them on paper and scan it. I'll edit this post later on, so just check it out.
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The next scribe is Ashlynn. I can't pick Madame President, Lindsay, because both of us, including katrin and linger won't be in Tuesday's class. Someone else pick her =).
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(Please ignore the dots =). I just put them cause when I publish or save, my whole post gets compressed; all the spaces are gone. So to make things clearer, I added these thingies. Sorry)