Evaluate $\displaystyle z_x$ at $\displaystyle (1, 2, 3)$ if $\displaystyle x^3+y^3+z^3+6xyz=72$

Part of me says I must solve this in terms of z in order to find $\displaystyle z_x$ but I can't see how.

The other part thinks I can just find $\displaystyle z_x$ implicitly and plug in the point but I am getting an answer that doesn't make any sense.

What should I do?