First of all, $g(x,y,z)= \{{(x^2+y+z),(x-3y+z^5)}\}$ is the constraint given, for which the total derivate has to be determined.
Btw the function given was $f(x,y,z)= x^4+y^2+z^3$ which is not a vector.
In my opinion I need to calculate $dg/dx + dg/dy + dg/dz$, but these 2 components confuse me because I either receive a vector or it doesn't work, but my professor received a single component equation system. I don't understand how he arrives at this single component equation system after using the Lagrangrange multiplier method.