Answer process:
1. The domain requires that xy is not equal to 0, so x is not equal to 0 and y is not equal to 0. So choose one
2.z=x^2 + y^3 = 1 + 2^3 = 9
3. Take y as a constant, and take the partial derivative of x to get ZX = 3x2Y+(Y- 1) * Y * E XY.
Substitute x = y = 1 into the above formula and get Zx( 1, 1) = 3+0 = 3.
4. Take X as a constant and take the partial derivative of Y to get ZY = (x-1) xcos (xy)+xcos (xy).
Substitute x = 1 and y = pi into the above formula to get Zy = 0+cos(pi) =-1.
5. The first derivative is:
Zx = 2x * y^3
Zy = x^2 * 3y^2
The second derivative is:
Zxx = 2 * y^3
Zyy = x^2 * 6y
Zxy = 2x * 3y^2 = 6xy^2