Implicit differentiation

Daniel123 · 2008-02-18 10:03:15

I was just looking at implicit differentiation.. and I was wondering why it's OK to differentiate all the terms in the equation and the equation still holds?

Identity · 2008-02-18 19:55:35

Because it's the exact same thing on both sides, so both sides will behave the exact same way. Then apply

etc and all terms will be differentiated.

Last edited by Identity (2008-02-18 19:56:06)

luca-deltodesco · 2008-02-18 20:16:17

there is no such thing as explicit differentiation really, when you do explicit differentiation you are really missing a step and making an assumption.

how do you go from

to

before implicit differentiation you are told that you differentiate the right side f(x) and you get dy/dx. but from implicit differentiation you can differentiate both sides with respect to x. implicit differenetiation means you can differentiate for example x = y^2. by differentiating the right side with respect to y, and multiply by dy/dx. i.e.

which writing it a bit differently you can see is just the chain rule :

basicly with implicit differentiation you can keep it as the same equation since you are doing the same thing to both sides of the equal sign, whereas before you could only treat it as a new equation, if you were instead of differentiating, multiply by 10. You couldn't exactly go through and multiply every 'x' term by 10, and not do the same to the 'y' terms, or it wouldn't be equal anymore.

Last edited by luca-deltodesco (2008-02-18 20:20:22)

Daniel123 · 2008-02-21 05:07:45

What if you had something like 2x = 10? If you differentiated both sides you would end up with 2 = 0?

Kurre · 2008-02-21 06:01:32

Daniel123 wrote:

What if you had something like 2x = 10? If you differentiated both sides you would end up with 2 = 0?

2x=10 -> x=5, which is not a function and therefor cant be differentiatet..?

Daniel123 · 2008-02-21 06:58:33

Silly me.

Thanks

Identity · 2008-02-21 20:24:16

Kurre wrote:

Daniel123 wrote:
What if you had something like 2x = 10? If you differentiated both sides you would end up with 2 = 0?
2x=10 -> x=5, which is not a function and therefor cant be differentiatet..?

Neither is

, but you can differentiate that with respect to x.

luca-deltodesco · 2008-02-21 20:31:58

The way i see it, you cannot differentiate 2x = 10, because the derivitive is the rate of change of one variable with respect to another. but here you only have 1 variable.

Last edited by luca-deltodesco (2008-02-21 20:38:18)

mathsyperson · 2008-02-22 06:52:25

Then how come if you have y = 2, then you can differentiate to get dy/dx = 0 with no problems?

Kurre · 2008-02-22 08:31:25

mathsyperson wrote:

Then how come if you have y = 2, then you can differentiate to get dy/dx = 0 with no problems?

y depends on x, such that y=2 for all x, it doesnt change, so the derivative in respect to x is 0. But in the previous example we had x=5, and thus we cant have any values on y. in your example the single variable (y) is a function, while we didnt treat x as a function in the example x=5. lets say we make x dependant on y, then we will get dx/dy=0, which makes sense....
i think...:o

Last edited by Kurre (2008-02-22 08:31:56)

luca-deltodesco · 2008-02-23 00:40:11

mathsyperson wrote:

Then how come if you have y = 2, then you can differentiate to get dy/dx = 0 with no problems?

because you have two variables, you have x. you can differentiate y = 2 with respect to another variable, but you cannot differentiat y = 2 with respect to y.

JaneFairfax · 2008-02-23 03:08:41

Implicit differentiation, generally defined, involves finding

when given a function of the form .

It can be shown (using the generalized chain rule) that

provided, of course, that

.

And so, folks, this is why you can implicitly differentiate y = 2 with respect to x but not 2x = 10 with respect to x. In the latter,

and so does not exist.

Well, of course it doesnt exist! The graph of 2x = 10 is a vertical straight line!

Last edited by JaneFairfax (2008-02-23 03:21:04)

Math Is Fun Forum

#1 2008-02-18 10:03:15

Implicit differentiation

#2 2008-02-18 19:55:35

Re: Implicit differentiation

#3 2008-02-18 20:16:17

Re: Implicit differentiation

#4 2008-02-21 05:07:45

Re: Implicit differentiation

#5 2008-02-21 06:01:32

Re: Implicit differentiation

#6 2008-02-21 06:58:33

Re: Implicit differentiation

#7 2008-02-21 20:24:16

Re: Implicit differentiation

#8 2008-02-21 20:31:58

Re: Implicit differentiation

#9 2008-02-22 06:52:25

Re: Implicit differentiation

#10 2008-02-22 08:31:25

Re: Implicit differentiation

#11 2008-02-23 00:40:11

Re: Implicit differentiation

#12 2008-02-23 03:08:41

Re: Implicit differentiation

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