To find the second derivative of U with respect to x, we need to take the derivative of the first derivative with respect to x.

Given U = xsin(xy) + ycos(xy)

First, let's find the first derivative of U with respect to x:
dU/dx = d/dx(xsin(xy)) + d/dx(ycos(xy))
dU/dx = sin(xy) + xcos(xy)y + cos(xy)*dy/dx

Now, let's find the second derivative of U with respect to x:
d^2U/dx^2 = d/dx(sin(xy) + xcos(xy)y + cos(xy)dy/dx)
d^2U/dx^2 = 0 + cos(xy)y - xsin(xy)y^2 + cos(xy)*d^2y/dx^2

Therefore, the second derivative of U with respect to x is:
d^2U/dx^2 = cos(xy)y - xsin(xy)y^2 + cos(xy)d^2y/dx^2