Subscripts in partial derivative notation

[chans]

Hi everyone,

We just started learning partial derivatives and I understand the f_x notation, but I'm confused when I'm asked for the value of f_xy. Does this mean multiply the two derivatives together?

For example:

What is f_xy when f(x,y) = (x+2y)ln(xy)

Thanks!

 

[micromass]

It means $(f_x)_y$. So it asks for the partial derivative in the y-direction of $f_x$.

For example, if $f(x,y)= x^2y+5y^2$, then $f_x(x,y) = 2xy$ and $f_{xy}(x,y) = 2x$.

 

[chans]

So basically I'll take the partial derivative with respect to x, then take the derivative of that with respect to y?

 

[micromass]

So basically I'll take the partial derivative with respect to x, then take the derivative of that with respect to y?

Yes.

It is a curious property that (under very mild conditions), this is the same as taking the partial derivative with respect to y and then with respect to x. So $f_{xy}=f_{yx}$.