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

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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).

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