Calculate the second order linear derivatives fxy, fyx for the given function f(x,y)=x^3+8xy

We'll begin with fxy.

fxy =
d^2f/dydx = [d(df/dx)]/dy

We'll differentiate with respect
to x:

fxy =
d[d(x^3+8xy)/dx]/dy

fxy = d(3x^2 +
8y)/dy

We'll differentiate with respect to
x:

fxy = 8

We'll calculate
fyx:

fyx = d^2f/dxdy =
[d(df/dy)]/dx

fyx =
d[d(x^3+8xy)/dy]/dx

fyx =
d[d(8x)]/dx

fyx =
8

So, the second order linear derivatives
are: fxy = 8 ; fyx = 8.