given $f(x,y,z)=e^{xyz^6}$
calculate $f_{xyz}$
$f_{x}= yz^6e^{xyz^6}$ using the chain rule and multiplying the derivative of the outisde times the inside
then
$f_{xy} = 5yz^6e^{xyz^6} + e^{xyz^6}xz^6*yz^6$
$=5yz^6e^{xyz^6}+xyz^{12}e^{xyz^6}$
$f_{xyz}= [30yz^5e^{xyz^6}+e^{xyz^6}6xyz^5]+[12xyz^{11}e^{xyz^6}+e^{xyz^6}6xyz^5\times xyz^{12}]$
$= 30yz^5e^{xyz^6}+6xyz^5e^{xyz^6}+12xyz^{11}e^{xyz^6}+6x^{2}y^{2}z^{17}e^{xyz^6}$
It wouldn't accept my answer even after making corrections to $f_{xy}$ as suggested and taking that partial derivative I obtained $6xz^5e^{xyz^6}+6xyz^{11}e^{xyz^6}+6x^2yz^{17}e^{xyz^6}$ It said this was still wrong. This question is asinine to begin with and some feed back from those supposed "math lovers" out there would be great.
OK, let's do this: $$ f(x, y, z) = e^{x, y, z^6} $$ By the chain rule, with $[xyz^6]_x = yz^6$ we get $$ f_x(x, y, z) = yz^6e^{xyz^6} $$ By product rule $yz^6\cdot e^{xyz^6}$ (and chain rule) we get $$ f_{xy}(x, y, z) = z^6\cdot e^{xyz^6} + yz^6\cdot xz^6e^{xyz^6} = (z^6 + xyz^{12})e^{xyz^6} $$ and finally, by the product rule (and chain rule) again $$ f_{xyz}(x, y, z) = (6z^5 + 12xyz^{11})e^{xyz^6} + (z^6 + xyz^{12})\cdot 6xyz^5e^{xyz^6}\\ = 6z^5(x^2y^2z^{12} + 3xyz^6 + 1)e^{xyz^6} $$