The "gradient of F" here is the following sum:
(partial of F with respect to x)i + (partial of F with respect to y)j + (partial of F with respect to z)k.
Please, use "^" to denote exponentiation: F(x, y, z) = x2yz − xyz3 should be presented as F(x, y, z) = x^2yz − xyz^3.
The partial of F with respect to x is 2xyz. That with respect to y is
x^2z. That with respect to z is -3xyz^2.
Thus, your gradient is
G(x,y,z) = 2xyz i + x^2z j + -3xyz^2 k.