# Proof Partial Derivative definition

## Main Question or Discussion Point

Hello, i'm trying to proof the partial derivative definition , how do i proof it ??

@f/ @x = lim h-->0 lim [f(a +h, b) - f(a, b)] / h

If possible , i'd like to seen all the calculations

Best regards

Related Differential Equations News on Phys.org
CompuChip
Homework Helper
If this is what you are trying to prove, then what is your definition of the partial derivative $$\frac{\partial f}{\partial x}$$?

Fredrik
Staff Emeritus
Gold Member
You don't prove a definition.

Delta2
Homework Helper
Gold Member
erm unless u mean the easy derivation where u start by $$\frac{\partial f}{\partial x}=lim_{x->a}\frac{f(x,b)-f(a,b)}{x-a}$$

you can use the function $$g(h)=a+h$$

and replace x=g(h) since $$(lim_{x->a}x)=a=(lim_{h->0}g(h))$$.

Last edited:
CompuChip
Homework Helper
Hello, i'm trying to proof the partial derivative definition
Ah, I missed that you are trying to prove the definition itself.

Well, that is very easy: by definition, it is true.

It was my Math teacher who challenged me to proof that the definition is right.

But i don't how to doi it.

I already proof the simple derivative definition.

Thanks

Fredrik
Staff Emeritus
Gold Member
It was my Math teacher who challenged me to proof that the definition is right.

But i don't how to doi it.

Note that you never prove a definition. A definition is just a choice of what English word to use for a specific mathematical concept.

HallsofIvy
Homework Helper
Perhaps your teacher meant this: suppose f(x, y) is, putatively, a function of x and y but does not, in fact, depend on y. Show that the partial derivative with respect to x is exactly the same, in this case, as the ordinary derivative (and that the partial derivative with respect to y is 0).

CompuChip
Homework Helper
Or it can mean: prove that it satisfies the Leibniz ("chain") rule.
Or that is satisfies a product rule.
Or both.
Or something else entirely.

It's just guessing here what your teacher meant, so maybe you should ask him to clarify :)

OK, i've just spoke with my teacher.
Now i've to proof that a function x^2 + y^2 is equal to 2x in order to x and by the definition ( limit ).

How do i do it ??

Thanks
Best Regards

Fredrik
Staff Emeritus
Gold Member
Do you know the definition of the derivative? Do you know the definition of a limit of a function? Show us your attempt to use these definitions, and we'll try to help.

I assume that what you're asking is either

"prove that if f(x)=x2+y2, then f'(x)=2x"

or

"prove that if "f(x,y)=x2+y2, then f,1(x,y)=2x".

(The ",1" notation is what I use for the derivative of f with respect to the first variable). The solutions to these problems will look almost exactly the same.

Last edited:
Delta2
Homework Helper
Gold Member
By using the definition you gave at the first post instead of a we use x and instead of b we use y and we have:

$$\frac{\partial f(x,y)}{\partial x}=lim_{h->0}\frac{(x+h)^2+y^2-(x^2+y^2}{h}=lim_{h->0}\frac{(x+h)^2-x^2}{h}=lim_{h->0}\frac{(x+h+x)(x+h-x)}{h}=$$
$$=lim_{h->0}\frac{(2x+h)h}{h}=lim_{h->0}(2x+h)=2x.$$

Now fredric is gonna be angry like his avatar image if he sees i gave u the full solution so get it fast :)...But i have to say maybe i did more harm to you than help by giving you the full solution.

Thanks Delta and Fredrik ....

I made a small mistake , when i did the indetermination 0/0. But when i saw the solution posted by Delta , i saw the error.

Thanks to you all.