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Differentiation/Integration using chain rule/reverse chain rule?

Hi, I have a few questions regarding when we can/can't use the chain rule/reverse chain rule to differentiate/integrate.

I know that you use them on functions that look like this: (ax + b)^n [is there a name for this type of function?]

And I know that you can only use chain rule if there is a linear function inside the brackets. But is that the only exception?

Is that the same for reverse chain rule? Does it only work for linear functions as well?

I'm just a bit confused when I can/can't use it :/ Also, say you had something like y=(2x+1)^2. Would you get all the marks if you wrote, dy/dx = 4(2x+1). Or would you need to do, let u=2x+1. Then y=u^2. dy/du = 2u. du/dx = 2. 2*2u = 4u = 4(2x+1)?

Thanks :smile:

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Reply 1
Avatar for Hasufel
Hasufel
16
you can use the chain rule for any composite function.

reverse chain rule will be for when your integrand is made up of a function of an essential function, and what is "essentially" it`s derivative, for example:

∫9x2(x3+1)2dx\int9x^{2}(x^3+1)^2dx (the x^3+1) is the essential function, and 9x^2 is "essentially" it`s derivative.

As for the way you present your work - i can`t comment on that, i`m afraid - i don`t really have any idea of the content of courses mentioned on tsr (sorry!)
(edited 11 years ago)
Reply 2
Avatar for StaedtlerNoris
StaedtlerNoris
OP
Original post by Hasufel
you can use the chain rule for any composite function.

reverse chain rule will be for when your integrand is made up of a function of an essential function, and what is "essentially" it`s derivative, for example:

∫9x2(x3+1)2dx\int9x^{2}(x^3+1)^2dx (the x^3+1) is the essential function, and 9x^2 is "essentially" it`s derivative.

As for the way you present your work - i can`t comment on that, i`m afraid - i don`t really have any idea of the content of courses mentioned on tsr (sorry!)


Ah, and a composite function is anything that is like fg(x)? So could you give me an example of a function which you can't use chain rule on? (Just so I can visualise it). I thought that you wouldn't be able to differentiate (2x^2+2)^3 by using the chain rule, but as that is a composite function, I guess you can?

So are there any functions where you can't integrate through using reverse chain rule?
Original post by StaedtlerNoris
(ax + b)^n [is there a name for this type of function?]


You could call it a power of a monomial :smile:
Original post by StaedtlerNoris
Ah, and a composite function is anything that is like fg(x)? So could you give me an example of a function which you can't use chain rule on? (Just so I can visualise it). I thought that you wouldn't be able to differentiate (2x^2+2)^3 by using the chain rule, but as that is a composite function, I guess you can?

So are there any functions where you can't integrate through using reverse chain rule?


There are functions you can't integrate using the 'reverse chain rule'.
One famous example is e-x2.
Reply 5
Avatar for Hasufel
Hasufel
16
yup.

you can`t use the chain rule on any function which is a product of 2 or more fuctions (unless one of them has an identical term)

e.g. can`t: y=(x2−1)3(x2+1)7y=(x^{2}-1)^3(x^{2}+1)^7

can: y=(x2−1)3sin(x)(x2−1)7y=(x^{2}-1)^3sin(x)(x^{2}-1)^7 (as this becomes: y=(x2−1)10sin(x)y=(x^{2}-1)^{10}sin(x)

you CAN differentiate (2x2+2)2(2x^2+2)^2 using the chain rule, because, the chain rule is for composite functions.

(it becomes: y=u2y=u^2 where u=2x^2+2.

There are many more types you just don`t use the chain rule on at all.

Basically, if the function y has a certain term which is having another (or more than one other) (EDIT) rule applied to it, you can use the chain rule.

As a point, a tricky example of this would be: sin(tan(x2)+1)sin(tan(x^2)+1)

the functions you can`t integrate using the reverse rule: firstly, they have to be a product to be able to use it in the first place (EDIT), and secondly, they have to have "essentially" related derivatives.

like the one above. The derivative for this is the derivative for each individual function:

the "innermost" function is x2x^2 whose derivative is 2x2x

the second innermost is tan()+1tan()+1 whose derivative is sec2()sec^{2}()

and the outermost is sin()sin() whose derivative is cos()cos()

giving:

2xsec2(x2)cos(tan(x2)+1)2xsec^{2}(x^2)cos(tan(x^2)+1)

(as a question, though, it`s quite hard at first glance to see what the integral of this is)
(edited 11 years ago)
Reply 6
Avatar for StaedtlerNoris
StaedtlerNoris
OP
Original post by placenta medicae talpae
There are functions you can't integrate using the 'reverse chain rule'.
One famous example is e-x2.


I think that's too complicated for me :tongue: I'm doing C3 (OCR board) at the moment :smile:

I've found a few examples where you can't use normal chain rule. Are these the only cases where chain rule doesn't work?

1. x (2x +1)^2
2. (2x+1) ^2x

Are these the only cases? (when there is an x in the power/in front of the function)

And are these exceptions the same for reverse chain rule?


I was differentiating a e/ln equation, could I just run this by you?

Which method is better for differentiating y=2^x?

1. lny = xln2
x= lny/ln2
dx/dy = 1/(yln2)
dy/dx=yln2

2. lny=xln2
lny=ln2^x
y=e^2^x
[Then I'm a bit stuck- is this wrong?]
Reply 7
Avatar for Haesights
Haesights
6
Original post by StaedtlerNoris
I think that's too complicated for me :tongue: I'm doing C3 (OCR board) at the moment :smile:

I've found a few examples where you can't use normal chain rule. Are these the only cases where chain rule doesn't work?

1. x (2x +1)^2
2. (2x+1) ^2x

Are these the only cases? (when there is an x in the power/in front of the function)

And are these exceptions the same for reverse chain rule?


I was differentiating a e/ln equation, could I just run this by you?

Which method is better for differentiating y=2^x?

1. lny = xln2
x= lny/ln2
dx/dy = 1/(yln2)
dy/dx=yln2

2. lny=xln2
lny=ln2^x
y=e^2^x
[Then I'm a bit stuck- is this wrong?]

It should be y=2^x if you cancel out the lns
Edit: what are you doing in the second method?
(edited 11 years ago)
Reply 8
Avatar for Hasufel
Hasufel
16
for the first one, you can use the chain rule on the second part of it.

as for the method for differentiating y=2^x, the standard derivative for dy/dx {a^x} = {a^x}ln(a) because a^x is equivalent to e^{xln(a)}

but, for the derivative of anything where x is in the power - for example
Unparseable latex formula:

y^x=a^{x}(2x+1)^



becomes(by taking logs) : xln(y)=xln(a)+ln(2x+1)xln(y)=xln(a)+ln(2x+1)

which is easy - if you know implicit differentiation.

always take natural logs first before differnetiating.
Reply 9
Avatar for StaedtlerNoris
StaedtlerNoris
OP
Original post by Haesights
It should be y=2^x if you cancel out the lns
Edit: what are you doing in the second method?


Can't you raise everything to the power of e? So e^lny = y, and e^ln2^x = 2^x? Oh wait, that just goes back to y=2^x :facepalm: Does the second method go anywhere? :redface:

Original post by Hasufel
for the first one, you can use the chain rule on the second part of it.

as for the method for differentiating y=2^x, the standard derivative for dy/dx {a^x} = {a^x}ln(a) because a^x is equivalent to e^{xln(a)}

but, for the derivative of anything where x is in the power - for example
Unparseable latex formula:

y^x=a^{x}(2x+1)^



becomes(by taking logs) : xln(y)=xln(a)+ln(2x+1)xln(y)=xln(a)+ln(2x+1)

which is easy - if you know implicit differentiation.

always take natural logs first before differnetiating.


Er I'm a bit confused :redface: I understood your last line, so that would suggest that method 1 is correct, and 2 isn't? We haven't done implicit differentiation yet, that may be the reason why I'm finding it confusing :/
Original post by Hasufel
yup.

you can`t use the chain rule on any function which is a product of 2 or more fuctions (unless one of them has an identical term)

e.g. can`t: y=(x2−1)3(x2+1)7y=(x^{2}-1)^3(x^{2}+1)^7



I do not really understand your post

You would use the Chain Rule when differentiating this function as it is a product of 2 composite functions
Original post by StaedtlerNoris



I've found a few examples where you can't use normal chain rule. Are these the only cases where chain rule doesn't work?

1. x (2x +1)^2



What do you mean by "normal" chain rule and by "these do not work"

This example is a product which will involve the use of the chain rule
Original post by TenOfThem
What do you mean by "normal" chain rule and by "these do not work"

This example is a product which will involve the use of the chain rule


I wasn't sure when you can/can't use the chain rule to differentiate something. I know that for that example, you would need to use the product rule, along with the chain rule. But are there any general expressions which the chain rule cannot be used to differentiate? Or conversely, when can you use the chain rule?
Reply 13
Avatar for Ursin
Ursin
14
Original post by placenta medicae talpae
There are functions you can't integrate using the 'reverse chain rule'.
One famous example is e-x2.


Can you not use it for any exponenetials? I tried it on my mock for a function containing ex and completely messed up the question :s-smilie:
Original post by StaedtlerNoris
I wasn't sure when you can/can't use the chain rule to differentiate something. I know that for that example, you would need to use the product rule, along with the chain rule. But are there any general expressions which the chain rule cannot be used to differentiate? Or conversely, when can you use the chain rule?


The chain rule is for any function of a function

This can be used alone or as part of the product or quotient rule
Original post by TenOfThem
The chain rule is for any function of a function

This can be used alone or as part of the product or quotient rule


Okay thanks. And the reverse chain rule is the same? As I seem to recall from class that there are different conditions for the chain rule and reverse chain rule? Does the reverse chain rule not work when there is an x^2 inside the brackets or something?
Original post by StaedtlerNoris
Okay thanks. And the reverse chain rule is the same? As I seem to recall from class that there are different conditions for the chain rule and reverse chain rule? Does the reverse chain rule not work when there is an x^2 inside the brackets or something?


The reverse chain rule works if a function (in a function) is multiplied by the derivative (ignoring constants as we can adjust for those)

Examples would be

x(x^2+3)^7

x^5(3x^6 - 2)^3

(2x+3)(x^2+3x-5)^6
Original post by TenOfThem
The reverse chain rule works if a function (in a function) is multiplied by the derivative (ignoring constants as we can adjust for those)

Examples would be

x(x^2+3)^7

x^5(3x^6 - 2)^3

(2x+3)(x^2+3x-5)^6


I've never noticed that :redface: So if the function is not multiplied by its derivative, then we can either multiply out the brackets to integrate, or we can't integrate? E.g. something like x(2x+3)^2
Original post by StaedtlerNoris
I've never noticed that :redface: So if the function is not multiplied by its derivative, then we can either multiply out the brackets to integrate, or we can't integrate? E.g. something like x(2x+3)^2


that about covers it

of course there are a number of methods we can use for integration, you will not have learnt them all yet

:biggrin:
Original post by TenOfThem
that about covers it

of course there are a number of methods we can use for integration, you will not have learnt them all yet

:biggrin:


Woo thanks :smile:

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