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24-09-2008: 24th September 2008 17:15
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#5
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Exalted and Worshipped Member
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Join Date: Nov 2005
Location: Guildford/Uxbridge
Posts: 920
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Re: C3 trig + differentiation revision help
C3 Differentiation is very easy if you know these 3 rules and when to use them (here I have inlcuded how I know them but you might want to learn them a bit more formally):
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CHAIN RULE: For when you have a function of x to the power of something, eg (sinx)^2 usually written sin^2x ('sin squared x').
"Bring the power down, multiply by f(x), then multiply by f'(x)"
Eg. Differentiate y=(sinx)^2
Bringing the power down gives you 2 at the front, then f(x)=sinx and f'(x)=cosx.....
dy/dx= 2.sinx.cosx
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PRODUCT RULE: When you have two functions of x multiplied by eachother.
"First times the differential of the second + the second times the differential of the first"
Eg. Differentiate y=sinx.sinx
dy/dx=sinx.cosx + sinx.cosx = 2.sinx.cosx
Which also agrees with our result above using the chain rule. (sinx.sinx=sin^2x)
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QUOTIENT RULE: When you have a function of x divided by another function of x.
"Bottom multiplied by the differential of the top - Top times the differential of the bottom ALL divided by the bottom squared. you may want to check this, I can't be bothered lol, it should be easy to check by doing a question.
Eg. Differentiate y=sinx/ (x^2+3x)
dy/dx={(x^2+3x).cosx - sinx.(2x+3)}/ {(x^2+3x)^2}
=(x+3)(x.cosx-2.sinx)/(x^4+6x^3+9x^2)
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Then if you get a really nasty question like
Find f'(x) where f(x)= x^2.e^2x.sin^2x/cos^3x
You can use a combination of the rules.
Eg Treat e^2x.sin^2x as a seperte part and work out the derviative of that using the product rule...then multiply it by x^2 and use the product rule on that expression. Then use the product rule on cos^3x knowing that cos^3x=cos^2x.cosx (use the product rule again on cos^2 as above to get the derivative of that). Then once you have the derivatives of the top and bottom use the quotient rule on the whole lot.
I.e break it up like so:
{x^2.[e^2x.sinx]}/{cos.x.[cos^2x]}
Hope this is clear, any problems just PM me!
Last edited by LeeC : 24-09-2008 at 17:43.
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