1. ## Find f_(xy)

Find $\displaystyle f_{(xy)}$ for $\displaystyle f(x,y)=\frac{4x^2}{y}+\frac{y^2}{2x}$

I did one with $\displaystyle f_{x}(x,y)\ for\ f(x,y)=e^{xy}(cos x\ sin x)$

and got $\displaystyle e^{xy}(sin x\ siny)$

Thanks

2. Originally Posted by JJ007
Find $\displaystyle f_{(xy)}$ for $\displaystyle f(x,y)=\frac{4x^2}{y}+\frac{y^2}{2x}$

I did one with $\displaystyle f_{x}(x,y)\ for\ f(x,y)=e^{xy}(cos x\ sin x)$

and got $\displaystyle e^{xy}(sin x\ siny)$

Thanks
Differentiate once w.r.t x, keeping everything else as constant. Then differentiate the results w.r.t y, keeping everything else as constant.

3. $\displaystyle f=\frac{4x^2}{y}+\frac{y^2}{2x}$

$\displaystyle f=4x^2y^{-1}+\frac{y^2x^{-1}}{2}$

Take derivative w.r.t $\displaystyle x$ of $\displaystyle f$

$\displaystyle f_x=8xy^{-1}+\frac{-y^2x^{-2}}{2}$

$\displaystyle f_x=8xy^{-1}+\frac{-y^2x^{-2}}{2}$

Take derivative w.r.t $\displaystyle y$ of $\displaystyle f_x$

$\displaystyle f_{xy}=-8xy^{-2}+\frac{-2yx^{-2}}{2}$

Anygood?

4. Originally Posted by pickslides
$\displaystyle f=\frac{4x^2}{y}+\frac{y^2}{2x}$

$\displaystyle f=4x^2y^{-1}+\frac{y^2x^{-1}}{2}$

Take derivative w.r.t $\displaystyle x$ of $\displaystyle f$

$\displaystyle f_x=8xy^{-1}+\frac{-y^2x^{-2}}{2}$

$\displaystyle f_x=8xy^{-1}+\frac{-y^2x^{-2}}{2}$

Take derivative w.r.t $\displaystyle y$ of $\displaystyle f_x$

$\displaystyle f_{xy}=-8xy^{-2}+\frac{-2yx^{-2}}{2}$

Anygood?
Perfect.

5. Originally Posted by Mush
Differentiate once w.r.t x, keeping everything else as constant. Then differentiate the results w.r.t y, keeping everything else as constant.
Ok got it. So if I wanted $\displaystyle f_{xx}(x,y)$ then I differentiate w.r.t.x twice right?
$\displaystyle f(x,y)=e^{x^2y}$
$\displaystyle f_{x}(x,y)= 2xye^{x^2y}$
$\displaystyle f_{xx}(x,y)=(4x^2+2y)e^{x^2y}$

6. Originally Posted by JJ007
Ok got it. So if I wanted $\displaystyle f_{xx}(x,y)$ then I differentiate w.r.t.x twice right?
$\displaystyle f(x,y)=e^{x^2y}$
$\displaystyle f_{x}(x,y)= 2xye^{x^2y}$
$\displaystyle f_{xx}(x,y)=(4x^2+2y)e^{x^2y}$
That's right.