What is the partial derivative fx of the function f(x,y)=x^2*2^(x*y)?

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^2*2^(x*y))/dx

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

d (x^2*2^(x*y))/dx = (x^2)'*2^(xy) +
x^2*(2^(xy))'

d (x^2*2^(x*y))/dx= 2x*2^(xy) + x^2*2^(xy)*ln
2*(xy)'

We'll factorize by
2^(xy):

d (x^2*2^(x*y))/dx= 2x*2^(xy) + x^2*2^(xy)*ln
2*(y)

d (x^2*2^(x*y))/dx= 2^(xy)*(2x +
y*x^2*ln2)

The partial derivative is: fx =
2^(xy)*(2x + y*x^2*ln2)