Verified answer

Holding y constant and differentiating with respect to x yields

6yz * ∂z/∂x - [4e^(4x) cos(4z) + e^(4x) * -4 sin(4z) ∂z/∂x] - 0 = 0

==> [6yz - 4e^(4x) sin(4z)] ∂z/∂x = 4e^(4x) cos(4z)

==> ∂z/∂x = 2e^(4x) cos(4z) / [3yz - 2e^(4x) sin(4z)].

Similarly, holding x constant and differentiating with respect to y yields

[3z^2 + 6yz ∂z/∂y] - e^(4x) * -4 sin(4z) ∂z/∂y] - 6y = 0

==> [6yz + 4e^(4x) sin(4z)] ∂z/∂y = 6y - 3z^2

==> ∂z/∂y = (6y - 3z^2) / [6yz + 4e^(4x) sin(4z)].

I hope this helps!