what does y with respect to x mean

guss

I exercise not understand the departure between taking the derivative, and taking the derivative with respect to 10, or taking the derivative with respect to y (or any other variable).

If I take the derivative of y = x^two, I get y' = 2x. What if I use the dy/dx or but the d/dx annotation?

so

dy/dx y = dy/dx x^2
vs
d/dx y = d/dx 10^ii

another example I don't understand would be

dy/dx = 2x
vs
d/dx = 2x
vs
f'(x) = 2x

I know that the d refers to an infinitesimally pocket-sized number, but I just don't understand the deviation between the stuff I mentioned before.

Someone enlighten me?

wisvuze

if you lot differentiate y(x^ii ) with respect to x, y'all become 2xy. if yous differentiate y(x^ii) with respect to y, you lot get ten^2. What is going on is that 1 is seen as a function of ten and the other is seen as a role of y. That is, when y(x^ii) is a role of y, after fixing some ten, ten^two is simply a constant, so that differentiating y(ten^2) is like differentiating cx with respect to x, resulting in c.

guss

Sorry, I notwithstanding don't understand.

gb7nash

Ok, so let'south assume you take a role y = f(x). There'south a lot of overlap in notation, as yous'll see:

f'(x) means to take the derivative of y with respect to 10. (same with y')

d/dx ways to accept the derivative of whatever's later on information technology with respect to 10. For case:

d/dx (y), would mean to take the derivative of y with respect to ten.

dy/dx ways to take the derivative of y with respect to x. The "numerator" indicates what role you're taking the derivative of. The "denominator" indicates what yous're differentiating with respect to.

micromass

I actually detest the d/dx notations and similar...

I do not empathise the difference betwixt taking the derivative, and taking the derivative with respect to ten, or taking the derivative with respect to y (or any other variable).
If I have the derivative of y = x^2, I get y' = 2x. What if I utilise the dy/dx or just the d/dx notation?

The bespeak is that y is actually a role, then it would exist better to write y(ten)=x^2. And then dy/dx just ways the derivative of y with respect to 10. So

[tex]\frac{dy}{dx}=y'[/tex]

If y'all want to evaluate this in the point 2, then you write

[tex]\frac{dy}{dx}(2)[/tex].

Sometimes, if y=x^2, for case, people will write

[tex]\frac{dx^2}{dx}[/tex] instead of [tex]\frac{dy}{dx}[/tex]

But I consider that to exist very bad notation...

so
dy/dx y = dy/dx x^two
vs
d/dx y = d/dx x^2

The first notation doesn't really makes sense to me. The 2d would be

[tex]\frac{d}{dx}y:=\frac{dy}{dx}=y'[/tex]

another example I don't sympathize would be
dy/dx = 2x
vs
d/dx = 2x
vs
f'(10) = 2x

The second notation doesn't make sense to me. The first does, but I remember it's bad notation and I would never use it...

I know that the d refers to an infinitesimally small number, but I just don't empathize the difference between the stuff I mentioned earlier.

Not everybody will concord with me, but don't call up of d equally infinitesimal number. Just call up of d as a notation. Thinking of d as a number causes you to make mistakes, and in (standard) real numbers, there are no such things as infinitesimals...

guss

Thanks guys, I think I'm starting to understand information technology.

The "denominator" indicates what you're differentiating with respect to.

I still don't understand what this means, though. What does "with respect to" actually mean?

micromass

Information technology'south nothing spectacular, "with respect to" simply indicates the variable.

For example, if f(10)=2x, then f'(x)=ii, and the notation would be df/dx
Just we can as well write f(z)=2z (this is the same function), then we would write df/dz.

This annotation is useful for functions like f(ten)=2a+10, where a is just a number. If nosotros do not know what our variable is (x in this example), then we could both have df/dx or df/da. The dx in the bottom just serves equally a reminder to what the variable of f is called...

guss

Ahh, I sympathise now. Thanks!

Simply, concluding question. In explicit differentiation, d/dx is usually used to represent the alter of the function with respect to 10. However, in implicit differentiation, why is dy/dx used to represent the modify of a part with respect to x?

jhae2.718

When you lot do implicit differentiation, y is a function of 10 so when you take the derivative of y with respect to ten y'all write it equally a derivative of the role.

When you differentiate an explicit office of 10 y'all know how the part is dependent on x so you tin can explicitly accept the derivative. Yous don't know how y depends on x, and so yous must go out it as dy/dx.

guss

I'grand not actually post-obit, sorry. I think we take a misunderstanding in your second paragraph. I am just referring to an equation like y = 5x^2 or f(x) = 5x^two. Not a multivariable expression.

jhae2.718

Ah, I thought you meant implicitly differentiating a function like xy^2 = 2x/y or similar.

I'm not quite sure what yous hateful and so by explicit and implicit differentiation.

As far as the notation does, d/dx is just a differential operator, meaning have the derivative w.r.t. x, where equally dy/dx applies the operator to some function y.

guss

That is what I mean.

Yous said

When you differentiate an explicit role of ten you know how the function is dependent on 10 then you lot can explicitly have the derivative. You don't know how y depends on 10, and so you must leave it every bit dy/dx.

It seems to me that you are talking about something like f(x) = x^2 + 6y.

Could yous rephrase what you said earlier? Sad for being unclear I am very new to this stuff haha.