![863385284hamburger-fast-food-french-fries-Favim.com-483020[1].jpg](https://i.imgur.com/4PAO12Z.jpg "863385284hamburger-fast-food-french-fries-Favim.com-483020[1].jpg")
If v'(x) = u(x), and y denotes some function, does v'(y) = y*u(x)?
>>8061752
Sorry, meant:
Does v'(y) = y*u(y)?
>>8061752
What is v? What u? What is x?
Does v' mean the derivative?
>>8061759
Yes.
https://en.wikipedia.org/wiki/Notation_for_differentiation#Lagrange.27s_notation
v and u are both functions
>>8061766
>v and u are both functions
I meant what were they exactly.
If you mean that this is general case for any functions v,u and y then I am sorry to tell you that even a toddler could construct a counter example.
That is why I supposed v and u were actually functions constrained in some way so that this general case would make sense.
>>8061752
>hamburger
>actually chicken sandwich
fug off
>>8061780
Those are any functions which satisfy the equalities. For example, in the OP, we can clearly see that u(x) is a function which is equal to the derivative of v(x).
>>8061752
If y is a function that uses x then no. It also needs to be derivated according to derivation rules.
If it's just a number or not a function that includes x then yes
this conversation has me wondering, what is the most general case in which a function can be said to be differentiable?
do the domain and codomain need to be uncountably infinite? or is it something simpler?
>>8061789
Yeah but so what?
That is not enough of a restriction to make any of this a true statement.
Also, v' might aswell just be another function because we get no information about what v is so who cares.
>>8061804
>what is the most general case in which a function can be said to be differentiable?
https://en.wikipedia.org/wiki/Differentiable_function#Differentiability_in_higher_dimensions
>>8061756
No.
If you differentiate with respect to y:
v'(y) = u(y)
If you differentiate with respect to x:
v'(y(x)) = y'(x) u(y(x))
>>8061905
I also want to add to my post that Lagrange's notation isn't always clear. I prefer Euler' notation (or Leibniz's), since those notations allow you to mention the variable you differentiate to.
[math]\text{D}_{y}^1v(y)=u(y) \text{ and }
\text{D}_{x}^1v(y)=u(y)\text{D}_{x}^1y[/math]
