Find patial derivatives fx, fy if f(x,y) is given by f(x,y)=x*e^xy.

We'll calculate the partial derivative fx, differentiating
the expression of f(x,y), with respect to x, assuming that y is a
constant.

fx = df/dx = d
(x*e^xy)/dx

Since it is about a product, we'll apply the
product rule:

d (x*e^xy)/dx = (x)'*e^xy +
x*(e^xy)'

d (x*e^xy)/dx = e^xy +
x*y*e^xy

fx = (1 +
x*y)*e^xy

Now, we'll determine the partial derivative fy,
differentiating the expression of f(x,y), with respect to y, assuming that x is a
constant.

fy = df/dy = d
(x*e^xy)/dy

d (x*e^xy)/dy = (x)'*e^xy +
x*(e^xy)'

Since x is a constant, (x)' =
0

d (x*e^xy)/dy = 0 +
x*x(e^xy)

fy =
x^2*(e^xy)

The partial derivatives are: fx =
(1 + x*y)*e^xy ; fy = x^2*(e^xy).