# Proof Partial Derivative definition

Jonnyquest
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

Homework Helper
If this is what you are trying to prove, then what is your definition of the partial derivative $$\frac{\partial f}{\partial x}$$?

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

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:
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.

Jonnyquest
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

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.

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).

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 :)

Jonnyquest
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

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:
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.

Jonnyquest
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.