Find fxy if f(x, y)=xye^(x^2*y).

Well...if you are referring to: ∂^{2}f/(∂y∂x) then you just take the partial derivatives one at a time:

First use the product rule:

∂f/∂x = ∂/∂x[ (xy) * (e^(x^{2}y)) ] = ∂/∂x[xy]*(e^(x^{2}y)) + ∂/∂x[e^(x^{2}y)]*(xy)

= ye^(x^{2}y) + 2xye^(x^{2}y)(xy) = e^(x^{2}y) * (y + 2x^{2}y^{2})

Then do the same process to ∂f/∂x, this time taking the partial derivative with respect to y:

∂^{2}f/(∂y∂x) = ∂/∂y[ ∂f/∂x ] = ∂/∂y [ e^(x^{2}y) * (y + 2x^{2}y^{2}) ] = ∂/∂y[ e^(x^{2}y) ] * (y +2x^{2}y^{2}) + ∂/∂y[ y + 2x^{2}y^{2} ] * e^(x^{2}y)

= e^(x^{2}y)*x^{2}*(y+2x^{2}y^{2}) + (1+4x^{2}y)*e^(x^{2}y) = e^(x^{2}y) * (x^{2}y + 2x^{4}y^{2} + 1 +4x^{2}y) = e^(x^{2}y) * (2x^{4}y^{2} + 5x^{2}y
+ 1)