Given function is f( x ) = e^2xy

we have to calculate fx, fy, fxx, fxy, fyx, fyy

these are first order and second order derivative with respects to x , y.

 

fx = ∂f / ∂x

= ∂( e^2xy ) / ∂x ( differntaition of e^ax = a e^ax and if a is a function of x its differentaition with respect to x should be done.)

= e^2xy * 2y

 

fy = ∂f / ∂y

= ∂( e^2xy ) / ∂y

= e^2xy * 2x

 

fxx = ∂²f / ∂²x

double differentiation of function with respect to x.

we know that fx = e^2xy * 2y

so fxx = 2y * e^2xy * 2y

fxx = 4y² e^2xy

 

 

fyy = ∂²f / ∂²y

double differentiation of function with respect to y.

we know that fy = e^2xy * 2x

so fyy = 2x * e^2xy * 2x

fyy = 4x² e^2xy

 

fxy = ∂²f / ∂x ∂y

differentaition of of fx with respect to y.

we know that fx = e^2xy * 2y

so fxy = using product rule

e^2xy 2 + 2y *e^2xy * (2x)

2e^2xy ( 1 + 2xy )

 

 

fyx = ∂²f / ∂y ∂x

differentiation of fy with respect to x

we know that fy = e^2xy * 2x ( using product rule)

so fyx = e^2xy (2) + 2x *e^2xy * (2y)

f_yx =  2e^2xy ( 1 + 2xy )