# Partial Differentiation

#### FizzyCrow

##### New member
I do not have a specific problem in mind, i just need to be told, in stupid terms hehehe, how to do a second partial differentiation. I know that there is the partial one with respect to x^2 and y^2, i an having problems understanding how you get the xy, yx one. I also know that they are, in my case, always equal. Any help is appreciated, if you need a problem, i can see if i can pull one from my book.

I also wanted to say thank you to everyone that helped me in my last posts, i've just been so busy, i have not been able to respond, so THANKS

#### Unco

##### Senior Member
$$\displaystyle f(x,y) = x^2 + y^2$$

$$\displaystyle f_x$$ means the partial derivative of f(x,y) with respect to x. As you know, we treat y as a constant and differentiate wrt x:

$$\displaystyle f_x = 2x$$
or $$\displaystyle \frac{\partial f}{\partial x} = 2x$$

Likewise,
$$\displaystyle f_y = 2y$$ or $$\displaystyle \frac{\partial f}{\partial y} = 2y$$

$$\displaystyle f_{xy}$$ reads from left to right in that: "the partial derivative of $$\displaystyle f_x$$ wrt y".

But with the chain rule reads:

$$\displaystyle \frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x}\right)$$ (which here = 0)