Maths bods - Differentiation

[J.M.]

I'm trying to get my head around differentiation at the moment. It's the last topic I have to understand before my test on Tuesday. In reality, it's not that important because I can choose 12/16 questions to do so I can just skip any questions on differentiation.

Anyway... I'm currently working through it the long way to give myself a fundamental understanding of how it works. I've made myself a sample problem of y = 12x^5 + 15. It's easy to see that the answer is 60x^4, but I'm working on the process of why. This is my workings out:

https://i.imgur.com/bprF8tJ.png?1

The thing I've highlighted in red is what bugs me. As delta x approaches 0, this fraction will become undefined, will it not? In which case we have dy/dx = 60x^4 + undefined, which isn't a valid answer.

Can anybody explain that part to me please?
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2004 R1 & 2018 XSR900

[Nick 50]

Can I ask, why you have the bit in red as a fraction??

Sure it is just 12(delta x)^4? and then as delta x approaches 0 it just disappears like the other elements containg delta x??
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Current Bikes: ZX7r 97 (Black Beauty), VFR400 NC24 (The banana)
Previous Bikes: Aprilia Tuono 03 (The Beast), CBR600f (97)
First bike: A GZ125 Lemon.......

[J.M.]

Bah of course! It's been a long day... somehow I was reading it as 128x^5 not 12 delta x to the 5.

That fixes it
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2004 R1 & 2018 XSR900

[Nick 50]

[J.M.]

[swampy]

I love lamp.....
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"I hate to advocate drugs, alcohol, violence or insanity to anyone, but they've always worked for me." Hunter S Thompson
"Faster, faster, faster, until the thrill of speed overcomes the fear of death..." Hunter S Thompson

[Nick 50]

[J.M.]

It's exactly the same for me I'm terrible at learning in class through listening. Especially with my tutor. He really knows his stuff, but he goes off on massive tangents. If you understand what he's going on about then it would be really interesting... if you don't understand the stuff before hand then it's like... EHHH?! I need diagrams and paper and then repeat until I figure it out and it sticks in my mind! But I can't just accept something works... I always want to know why it works too!

The best maths teacher I've ever had was back studying Sats in year 9. I could have done with him this year!

I suspect you're probably studying slightly different things to me, but thank you for the offer

I've currently got this book:
https://ecx.images-amazon.com/images/I/51Q%2BTJR7DgL.jpg

It's pretty darn amazing, if you don't have it I would really recommend it! It has explained 99% of things to me in a way which I can understand them. Then Khan Academy usually manages to explain the rest to me.

There are just some things that I am still lost on with Trigonometry:
Things like why:
Cos(theta + phi) = cos(theta)cos(phi) - sin(theta)sin(phi)

I get how to use them... but can't for the life of me figure out how it is derived!
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2004 R1 & 2018 XSR900

[swampy]

[J.M.]

[D O G]

No, it's in Portsmouth, it is an excellent thing that you don't recognise it.

[J.M.]

Visible from the ferry to the Isle of Wight... right... I do feel stupid sometimes. I've been on that ferry 4 times.

https://sphotos-e.ak.fbcdn.net/hphotos-ak-ash4/306221_381174671917502_28572408_n.jpg
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2004 R1 & 2018 XSR900

[spaz]

Ugh, calculus.


*barf*


Haven't used it for the last 6 years and don't intend to use it ever again!
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"If it's got tits or an engine, it'll be trouble"

[Rogerborg]

In my first job as a games developer, our resident maths nerd was doing ballistics calculations with all sorts of integrate-the-polynomial-of-the-arctangents-of-your-age stuff to try and work out the right angle to fire a mortar to have it land at the desired position under a variety of conditions. He'd been at it for days, with much Aspie mutterings of "...nearly there, just need to carry the i..."

I wrote a "pick an angle, run the simulator, see where it would land, iterate and binary chop until it's close enough" bodge in 20 minutes, committed it, and got on with the next problem.

In the real world, 4 thou is usually close enough.
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Biking is 1/20th as dangerous as horse riding.
GONE: HN125-8, LF-250B, GPz 305, GPZ 500S, Burgman 400 // RIDING: F650GS (800 twin), Royal Enfield Bullet Electra 500 AVL, Ninja 250R because racebike

[Lord Percy]

That method using oX and oY is a little strange. I've seen it before but I hate it. Here's an easier way

edit: Looks like you did it all properly anyway! That bit where your're dividing by oX, don't worry, it just becomes zero, not infinity. I'll leave the rest of the post here anyway because.... just because.
edit edit: Actually my method is a tiny bit different. It goes straight into the dy/dx, instead of mucking about with that y+Oy=x+Ox malarkey.

So the gradient of a line is (change in y) / (change in x)
The childish way to find the gradient is to draw a triangle and look at dy/dx. Obviously with a cruved line, the triangle will not be a very accurate form, unless it's very very small. Calculus basically considers how to push the triangle to it's smallest size such that it disappears and therefore the calculated gradient is perfect. So anyway...

We know y = 12x^5 +15

So the change in y is caused by an change in x. Call this change h.
Here it is in 'triangle' form.

https://cdn.bikechatforums.com/files/calc2.jpg

Now we can see:

Change in y: dy = [12(x+h)^5 + 15] - [12x^5 + 15]
Change in x: dx = (x+h)-x = h

Now write it in the form of dy/dx
Simplify it.
And consider what happens as h gets smaller and smaller, as it tends to zero.
See below

https://www.bikechatforums.com/download.php?id=89080

[J.M.]

Slightly confused looking at that way of doing things, but that's probably because my maths skills aren't brilliant

Differentiation like the question I posted above is really nice, because you can skip all of the workings out and go directly to the answer:
https://i.imgur.com/VvU8qa8.png

Now I just need to practise the things like the chain rule/product rule and run through some past papers and I'm ready for my test on Tuesday!
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2004 R1 & 2018 XSR900

[Lord Percy]

the method I wrote was exactly the same as yours, only I used 'h' where you used 'ox'. Although one thing that does seem odd, is the bit line where you 'subtract y'. You take y from one side of the equation but it doesn't seem logical to magically remove it for no reason.

I tried to understand all the fundamentals of everything when I first started learning maths. At some point I just stopped because memorising it seemed easier, but now somehow I can go back to the stuff I learned earlier and am able to show the proofs of how they work, even though at the time I didn't know . I guess things just click when you give them time to sink in.

good luck bud !!

Just wait till you have to use differential equations to resolve variable forces . Me and Smegballs had fun with that. (NB - Smegballs chage your username, anyone who mentions it sounds gay/perverse )

[J.M.]

We subtract y because we wish to isolate the equation in terms of δy and δx, meaning we can then approach δx to 0 to find dy/dx. At least, that's the reason I can find for subtracting y.

Just looked back at you post and figured out the part which was confusing me... now I feel stupid. I will blame it on being awake too early!

Fortunately I doubt that I'll ever have to use differential equations to resolve variable forces; I'm not doing Physics! Merely Computer Science/Software Engineering with no intentions to go in to games development. So with any luck, stuff like that shouldn't cross my path, ever
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2004 R1 & 2018 XSR900

[Lurkio]

[J.M.]

Yeah that is my plan

Currently I just need to study the things so I can score enough to get me on to the 2nd year of my course. Then over summer I can improve my math skills in preparation for the maths we are studying next year. I'm not sure what we are covering next year, but it has been very apparent this year that I have forgotten most things I've learnt up until now in maths!
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2004 R1 & 2018 XSR900

[P.addy]

[Derivative]

The reasoning behind 'as it goes to zero' is exactly that.

Differentiation and integration use infinitesimals (infinitely small numbers). Not zero - that's the key point.

0.1 / 0.1 is defined.
0.01 / 0.01 is defined.

That's what 'in the limit' means.

For example, let's look at the limits of (1+x)/(1-x).

Let's pick an arbitrary value of x, say, 0.1.
That gives 1.1/0.9.
If x is 0.01, that gives 1.01/0.99.
This is clearly converging to 1/1, as expected if you just actually set x = 0.

In the high limit: let's try x = 100.
101/-99. Try a few more and you'll see that this is clearly converging to -1.

https://en.wikipedia.org/wiki/Limit_%28mathematics%29 has some more formal information.

However - to be very frank - if you want to know how maths works, you should be doing a Mathematics degree. You simply don't have time otherwise. I'm doing Physics which is about as close as you can get to maths without studying it formally, and we have absolutely no chance of deriving everything we need. You have to let go of 'how do we formally prove this' at some point.

[J.M.]

Limits is one of the things I will be studying. I understand a small amount about them, but not enough yet.

I'm not all too fussed on how maths works. I'm here for the Computer Science degree with plans to go in to Software Engineering and frankly I'm scoring some of the top results in my year and I'm not great at maths. I just have to pass the module to be allowed to continue the course. I just like to understand "why" the things work which I need to know. At no point do I intend to fully understand the entirety of mathematics I find that if I understand "why" something works, I remember exactly how to do it because the "easy methods" we're taught all make perfect sense.

The reason I wish to study over the summer is because I have to study maths next year as a module too. So I need to become more comfortable with the processes that will be the foundations of what is studied next year. I.E. everything studied this year.

I wrote out something like 25 pages of notes for one of my modules summarising all of the processes behind the topics covered. Exam was a piece of cake as a result; because understanding "why" allowed me to describe the processes in the exam exactly how they worked rather than simply reiterating what the lecturer wrote in his notes.

@Paddy... 1.5 beers. You're liver's fucked. Dibs on the pizza once you're holding on to the floor so that you don't fall off the boat.
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2004 R1 & 2018 XSR900

[Cunnington]

[J.M.]

The only one use for differentiation thus far has been in Artificial Intelligence as part of machine learning. I.E. classifying/fitting data points. Although we were just given the algorithm to use for that and I suffered no issues.

But yeah "obviously" in Maths annoys me too. "So obviously xys is potato".

I'm going to have to just try and memorise that line that you quoted. And the other similar ones like 2nd angles in Trig. Differentiation seems to use them as does the dest paper. Oh... must remember the outputs of differentiating the sin/cos/tan/cosec/etc so I don't have to work them out in the exam. Saving time...

2 days to go... hope this goes well. It's the only exam I've really been worried about. Then 9 days to revise for my final 2 exams. Then I'm freeeeeee!
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2004 R1 & 2018 XSR900