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#1 2005-12-14 04:31:21

Tredici
Member
Registered: 2005-12-12
Posts: 28

Differentiation.

I've had a good look at all the existing topics based around this rather annoying subject, but nothing seems to satisfyingly relate to what I'm questioning. I don't quite understand how special functions react to differentiation, well I suppose I do, if basic differentiation. I'm just really having problems with questions involving the differentiation of multiple functions that are multiplied or squared.

There's a heck of a lot of questions I'm having problems with, and I'd much rather learn how to solve them as oppose to watching someone else work through it for me. So, I was thinking, would it be possible if someone could fill out a list of how the functions are differentiated, maybe if I were to provide a structure? At least, I know this:

f(x)          f'(x)   

sin(x)          cos(x)
cos(x)      -sin(x)
tan(x)         sec²(x)
sec(x)         sec(x)tan(x)
cosec(x)   -cosec(x)cot(x)
cot(x)          -cosec²(x)
ln(x)           1/x
e^x          e^x



I know, it's not much at all, but I'm just quite unsure of the following:




f(x)              f'(x)   

sin(ax)         
cos(ax)       
tan(ax)         
sec(ax)         
cosec(ax)     
cot(ax)         
ln(ax)             
e^(ax)         



f(x)            f'(x)   

sin²(x)       
cos²(x)       
tan²(x)       
sec²(x)         
cosec²(x)   
cot²(x)             
ln²(x)           





f(x)               f'(x)   

a x sin(x)         
a x cos(x)         
a x tan(x)         
a x sec(x)         
a x cosec(x)   
a x cot(x)         
a x ln(x)           
a x e^(x)       


and finally, just as an example:



f(x)                          f'(x)   

a x sin(bx + c)     
a x cos(bx + c)         
a x tan(bx + c)         
a x sec(bx + c)         
a x cosec(bx + c)     
a x cot(bx + c)         
a x ln(bx + c)               
a x e^(bx + c)         



I'd be eternally greatful if someone could show me how the above are differentiated. I understand it's an extensive list, maybe excessive, so it's perfectly understandable if you don't have the time for me, but I'm just eager to get to grips with this, as to cease resorting to this comunity with similar questions. Thanks so much for your time, I really do appreciate it. David.

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#2 2005-12-14 05:59:52

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Differentiation.

Each of the sections use the same method, so I'll show you how to differentiate the first of each one.

f(x) = sin (ax)

For this one, you need to use the chain rule: dy/dx = dy/dt * dt/dx.
Substitute t = ax.

dy/dt = cos t, dt/dx = a, ∴ dy/dx = a cos t = a cos (ax)
You'll find that that result is the same for most of them, but not all, so be careful.
For example, ln (ax) differentiated is still 1/x.
---
I'll come back to sin² x, because it's trickier than the rest.

a sin x, by contrast, is easy. Just take out the constant a and differentiate as normal, before putting the constant back on.

d(a sin x)/dx = a * d(sin x)/dx = a cos x.
---
a sin (bx + c) is just a combination of the above two things, and you should get a result of ab cos (bx + c).
---
As I said, sin² x is a bit trickier. For things like this, you need the product rule.
d(uv)/dx, where u and v are functions of x, = u*dv/dx + v*du/dx.

In this case, u = v = sin x.

d(sin² x)/dx = 2*sinx*d(sinx)/dx = 2sin x cos x.
---
I hope I helped.


Why did the vector cross the road?
It wanted to be normal.

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#3 2005-12-14 06:13:47

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Differentiation.

sin(ax)

The chain rule is most convenient for this one. (If you haven't learned it, the chain rule is a rule for differentiating composed functions. If ƒ(x) = sinx and g(x) = ax, then sin(ax) = ƒ(g(x)). The chain rule goes like this: (ƒ(g(x)))′ = ƒ(g(x))′ * g′(x).)

So. (sinx)′ = cosx, and (ax)′ = a. Plug these into the chain rule, and out pops a*cos(ax).

The same principle applies to the rest of the functions in that list. Another example is (ln(ax))′ = a/ax = 1/x.

sin²(x)

Rewrite this one as (sinx)(sinx). You can then use the product rule ((ƒ(x)g(x))′ = ƒ(x)g′(x) + ƒ′(x)g(x)).

(sin²(x))′ = sinxcosx + cosxsinx = 2sinxcosx (there's probably and identity to simplify that further, but I can never remember those darn things).

Again, the same idea applies to the rest of the list.

ax*sin(x)
Product rule again. ax*cosx + a*sinx.
Or did you mean to say a*sin(x)? That's just a*cos(x). Remember that (aƒ(x))′ = a(ƒ′(x))

a x sin(bx + c)

This one's an application of chain and product rules both.
sin(bx + c)′ = b*cos(bx + c)
(ax*sin(bx+c))′ = abx*cos(bx+c) + a*sin(bx+c)

I hope that helps you. If you were looking more for explanations of why the various rules works, there are some good proofs in my calculus book...


El que pega primero pega dos veces.

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#4 2005-12-14 06:15:03

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Differentiation.

Arrrgh, mathsy be a quick one. wink


El que pega primero pega dos veces.

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#5 2005-12-14 06:59:38

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Differentiation.

Indeed I be. Yarr.

ryos wrote:

(sin²(x))′ = sinxcosx + cosxsinx = 2sinxcosx (there's probably and identity to simplify that further, but I can never remember those darn things).

2 sinxcosx sin 2x


Why did the vector cross the road?
It wanted to be normal.

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#6 2005-12-14 10:01:43

Tredici
Member
Registered: 2005-12-12
Posts: 28

Re: Differentiation.

Wow, this is incredible. I like this community, such drive to offer a helping hand smile. This has actually been truly helpful. I understand almost all that you've said, apart from the use of 't' in the chain rule, that's confused me, I don't think we use the letter 't'. Anyway, I've got some fantastic questions involving differenciating that I'm hopefully about to power through, with your help that is smile. I shall report back to this thread should my limited knowledge fail me smile.

Thanks guys! smile

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#7 2005-12-14 10:06:51

Tredici
Member
Registered: 2005-12-12
Posts: 28

Re: Differentiation.

Oh guys I've just noticed, I haven't made myself too clear; I've written a x sin(x) when I meant more along the lines of (a)times[sin(x)]. How would that be differenciated?

Edit: Don't worry, I got it smile. Silly me.

Last edited by Tredici (2005-12-14 10:14:46)

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#8 2005-12-14 10:06:59

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Differentiation.

You don't have to use 't', that's just how we were taught. t is just a function of x, chosen to make the expression easier to differentiate.


Why did the vector cross the road?
It wanted to be normal.

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#9 2005-12-14 10:48:43

Tredici
Member
Registered: 2005-12-12
Posts: 28

Re: Differentiation.

Ok, and I'm back again.




ƒ(x) = 3 sin ² x + sec 2x

Ok, I'm not sure if this is correct, but I made that out to be:

(3 sin ² x)' = using chain rule, 3 [ 2 (sin x) (cos x) ] = 6 sin x cos x

(sec 2x)' = 2 sec (2x) tan (2x)

∴  ƒ'(x) = 6 sin x cos x + 2 sec (2x) tan (2x)

Please correct me if I'm wrong smile.





Another question.

[x + ln(2x)] ³

I'm very unsure of whether this is correct, but I ended up with:

[x + ln(2x)]' = (1 + 1/x)

using chainrule I ended up with:

ƒ'(x) = 3 [x + ln(2x)] ² (1 + 1/x)

Ok, looking back on that, it looks very wrong. So corrections would never go a miss.




Final One:

f(x) = 3e^x - ½ ln x - 2

Hmm... let's begin

(3e^x)' = 3e^x, right?

(½ ln x)' =  ½ 1/x  =  1/2x, right?

(2)' = 0

∴  ƒ'(x) = 3e^x - 1/2x.



Hmmm.. please let me know how wrong I am on these, lol. Only way to improve, eh?

Thanks again guys, as I've said before, really appreciate what you guys do, I'm surprised you don't charge!

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#10 2005-12-14 11:13:10

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Differentiation.

3/3, well done.  They are all correct.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#11 2005-12-14 12:07:11

Tredici
Member
Registered: 2005-12-12
Posts: 28

Re: Differentiation.

3? Out of 3? Correct? yikes.

Lmao, that's crazy, wouldn't of done it without you guys.

Let's suppose the last question was:

f(x) = 3e^x - ½ ln (x - 2) as oppose to f(x) = 3e^x - ½ ln x - 2

Would it then be..

(3e^x)' = 3e^x

(½ ln x - 2)' =  is that just, ½ 1/(x-2)? or simplified 1/(2x-4)

∴  ƒ'(x) = 3e^x - 1/(2x-4). Not quite so sure on that one, to be honest.

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#12 2005-12-14 12:55:51

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Differentiation.

Right again.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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