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Differentiation/integration - HSN forum

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Differentiation/integration


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#1 Daniel Williamson

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Posted 27 March 2006 - 07:51 PM

Any easy ways to remember the difference between the two? add one to power divide by new power i mean come on ph34r.gif

#2 The Wedge Effect

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Posted 27 March 2006 - 08:02 PM

Yeah. Practise plenty of questions.

Differentiation is when the power is reduced, at higher level you only need to know basic differentiation and trig.



\begin{align*} \frac {d}{dx} \left(x^n\right) = nx^{n-1}\,.\end{align*}

While in integration you raise the power by one and add an arbitary constant.



\begin{align*} \int {ax^n}\,dx= \frac {ax^{n+1}}{n+1} + C\,.\end{align*}

I believe derivatives and integrals of trig functions are given on the formula sheet in the Higher exam, which should help.

For the chain rule (if you've covered that), differentiating gives:



\begin{align*} \frac {d}{dx} \left((ax+b)^n\right) = an(ax+b)^{n-1}\,.\end{align*}

(not sure if this is right, need to double check with someone else.

Integrating gives:



\begin{align*}\int {(ax+b)^n}\,dx=\frac {(ax+b)^{n+1}}{a(n+1)}\,.\end{align*}


Also, consult the HSN notes on integrating and differentiating, if you need help.

Differentiation

Integration

Further Calculus

These notes provided on this website are excellent, providing many worked examples to go along with them. What you should do is pracise plenty of integration and differentiation questions until you get the hang of which is which. smile.gif

#3 Nathan

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Posted 08 April 2006 - 12:33 PM

bump

jus noticed this...one of the teachers in my school came up with a lame method, but anyway

Integration
Add
Divide
Differentiation
Multiply
Subtract

dunno how you would remember that in the exam tho :S

#4 The Wedge Effect

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Posted 08 April 2006 - 12:52 PM

Yeah, that sure is lame. I just practise lots of questions. Much easier than trying to remember that integration is addition then division, especially when you get to advanced higher level integration and you get stuff like:



\begin{align*}\int {\frac{1}{x}}\,dx=\ln x\end{align*}

#5 jackbauer

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Posted 09 April 2006 - 08:45 PM

QUOTE(nathanm @ Apr 8 2006, 01:33 PM) View Post

dunno how you would remember that in the exam tho :S


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lame, but it works for me!

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#6 SncZ

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Posted 09 April 2006 - 10:13 PM

sorry,maybe its just me, but i dont understand how that can help you learn ?

i have just done enough of them to know what to do when i see one, i think its the safest way to do it, rather than learning some little saying....


#7 Dave

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Posted 09 April 2006 - 11:20 PM

i always saw differentiation as the process of creating the mess (hence the easy process) and integration as the process of undoing the mess (hence the more complex process)

probably doesnt help you but it give everyone an insight into how my head is wired...scarey eh

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#8 Nathan

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Posted 10 April 2006 - 01:45 AM

QUOTE(Dave @ Apr 10 2006, 12:20 AM) View Post


probably doesnt help you but it give everyone an insight into how my head is wired...scarey eh


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#9 Daniel Williamson

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Posted 10 April 2006 - 01:15 PM

well thanx everyone really heres me looking for an easy answer and i have to practise great ph34r.gif

#10 Mikey_Pee

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Posted 10 April 2006 - 02:48 PM

A tip for you:

There are no easy answers in Higher Maths, unless you practice or are a natural mathematician.

I really don't like maths, but you just have to practice. See it as a month until you never have to do maths again in your life! That's keeping me going!

#11 Daniel Williamson

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Posted 13 April 2006 - 05:02 PM

SERIOUSLY NO EASY ANSWERS?? well i quit as of now
ph34r.gif

#12 Dave

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Posted 13 April 2006 - 05:44 PM

if it was easy everyone would be doing it

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#13 Nathan

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Posted 13 April 2006 - 06:31 PM

QUOTE(Mikey_Pee @ Apr 10 2006, 03:48 PM) View Post

A tip for you:

There are no easy answers in Higher Maths, unless you practice or are a natural mathematician.

I really don't like maths, but you just have to practice. See it as a month until you never have to do maths again in your life! That's keeping me going!


Unless you're foolish enough (like me) to take AH maths dry.gif

#14 Emma_P

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Posted 20 April 2006 - 05:27 PM

oohh thats harsh...why do that to yourself?? lol...unless maths is the way for you blink.gif

#15 dfx

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Posted 20 April 2006 - 06:48 PM

You should differentiate from first principles. Oh aye. Take 5 hours to do the Higher math exam but that way you understand what you're doing.

#16 ad absurdum

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Posted 20 April 2006 - 09:05 PM

QUOTE(dfx @ Apr 20 2006, 07:48 PM) View Post

You should differentiate from first principles. Oh aye. Take 5 hours to do the Higher math exam but that way you understand what you're doing.
Maybe not doing it in the exam, but at least knowing how to do it would definately help. Alot of people in my class don't know what differentiating a function actually does, but they have just memorised how to do it and how to use this. I think it's much more useful to know exactly what it is, and if a person can be told the first couple of steps then work this out using first principles then I think this is a great way to learn more about what differentiation actually does.
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#17 dfx

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Posted 20 April 2006 - 09:39 PM

I agree completely. Even in the Further Calculus bit I was taught to simply "multiply by the power and the number inside the bracket and drop the power" for further differentiation. They should teach the chain rule, or atleast mention the fact that you're multiplying by the derivative of the inner bracket.

#18 George

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Posted 20 April 2006 - 10:00 PM

QUOTE(dfx @ Apr 20 2006, 10:39 PM) View Post

They should teach the chain rule, or atleast mention the fact that you're multiplying by the derivative of the inner bracket.

Definitely, but most of the time it's sufficient to use the 'special version' of the chain rule, i.e. 

\begin{align*}\frac{d}{dx} (ax+b)^n = an(ax+b)^{n-1}\end{align*}

.

I would always read the a as the "derivative of the bracket" though.

I'm not so sure about differentiation from first principles. It is good to introduce, and have a go at, but - particularly at Higher - you can only take it so far because most of the time, the computations are far too involved (have you ever tried it with sinx? smile.gif ) That's the very reason why we have things like the chain rule!

#19 ad absurdum

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Posted 21 April 2006 - 12:18 PM

QUOTE(George @ Apr 20 2006, 11:00 PM) View Post

QUOTE(dfx @ Apr 20 2006, 10:39 PM) View Post

They should teach the chain rule, or atleast mention the fact that you're multiplying by the derivative of the inner bracket.

Definitely, but most of the time it's sufficient to use the 'special version' of the chain rule, i.e. 

\begin{align*}\frac{d}{dx} (ax+b)^n = an(ax+b)^{n-1}\end{align*}

.

I would always read the a as the "derivative of the bracket" though.

I'm not so sure about differentiation from first principles. It is good to introduce, and have a go at, but - particularly at Higher - you can only take it so far because most of the time, the computations are far too involved (have you ever tried it with sinx? smile.gif ) That's the very reason why we have things like the chain rule!
Yeah I see what you mean. I tried sinx just now and I have to admit I had to search google for a proof when I was calculating the limits. I still think that doing normal functions from first principles is a useful thing to be able to do though, and not too hard for the higher course (apparently it used to be in it).

I've never used that 

\begin{align*}\frac{d}{dx} (ax+b)^n = an(ax+b)^{n-1}\end{align*}

formula before, I always substituted part o the function with another function u and said 

\begin{align*}\frac{d}{dx} = \frac{d}{du} . \frac{du}{dx}\end{align*}

The rest of my class does it that way but I was teaching myself at this point and this was the way they done it in the textbook, I guess its better for me because it seems like less of a pointless operation and I think you can see the reasoning behind it better.

By the way, does anyone know why the stuff from the code seems to have a bit chopped off at the bottom?
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#20 dfx

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Posted 21 April 2006 - 02:02 PM

QUOTE(George @ Apr 20 2006, 11:00 PM) View Post

(have you ever tried it with sinx? smile.gif )


Oh yes I was recently challenged by my maths teacher to differentiate from first principles. I believe it involves the taylor and maclaurin expansions for Sinx... but yeah I gave up lol.





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