Please find fxy?

Question

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

Answer 1

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

First use the product rule:

∂f/∂x = ∂/∂x[  (xy) * (e^(x²y)) ] = ∂/∂x[xy]*(e^(x²y)) + ∂/∂x[e^(x²y)]*(xy)

= ye^(x²y) + 2xye^(x²y)(xy) = e^(x²y) * (y + 2x²y²)

 

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

 

∂²f/(∂y∂x) = ∂/∂y[ ∂f/∂x ] = ∂/∂y [ e^(x²y) * (y + 2x²y²) ] = ∂/∂y[ e^(x²y) ] * (y +2x²y²)    +  ∂/∂y[ y + 2x²y² ] * e^(x²y)

= e^(x²y)*x²*(y+2x²y²) + (1+4x²y)*e^(x²y) = e^(x²y) * (x²y + 2x⁴y² + 1 +4x²y) = e^(x²y) * (2x⁴y² + 5x²y + 1)