- #1

Moridin

- 692

- 3

## Homework Statement

Show through direct application of definition that the function

[tex]f(x,y) = xy[/tex]

is differentiable at (1,1)

## The Attempt at a Solution

I know that all functions of the class C

^{1}are differentiable and that a function is of the class C

^{1}if its partial derivatives exists and are continuous.

So do I prove that the two partial derivatives exist by showing that the following to limits exists?

[tex]\frac{\partial f}{\partial x}(a,b) = \lim_{h \to 0} \frac{f(a+h,b) - f(a,b)}{h}[/tex]

[tex]\frac{\partial f}{\partial y}(a,b) = \lim_{k \to 0} \frac{f(a,b+k) - f(a,b)}{k}[/tex]

The, the following obtains:

[tex]\frac{\partial f}{\partial x}(1,1) = \lim_{h \to 0} \frac{f(1+h,1) - f(1,1)}{h} = \lim_{h \to 0} \frac{(1+h)1 - 1}{h} = \lim_{h \to 0}\frac{h}{h} = 1[/tex]

[tex]\frac{\partial f}{\partial y}(1,1) = \lim_{k \to 0} \frac{f(1,1+k) - f(1,1)}{k} = \lim_{k \to 0} \frac{(1(1+k) - 1}{k} = \lim_{k \to 0}\frac{k}{k} = 1[/tex]

Since both partial derivatives are elementary functions, they are continuous, correct? If so, this ought to show that the function is differentiable in (1,1)?

Thank you for your time. Have a nice day.