Answer
$f_{xyz}=2ze^{xyz^{2}}(1+3xyz^{2}+x^{2}y^{2}z^{4})$
Work Step by Step
Consider the function $f(x,y,z)=e^{xyz^{2}}$
Let us start by finding $f_{x}(x,y,z)$ by differentiating $f(x,y,z) $ with respect to $x$ keeping $y$ and $z$ constant.
As we know
$f_{x}=\frac{∂}{∂x}f(x,y,z) $
$=\frac{∂}{∂y}[e^{xyz^{2}}]$
$=yz^{2}e^{xyz^{2}}$
Now, differentiate $f_{x}(x,y,z)$ with respect to $y$ keeping $x$ and $z$ constant .
$f_{xy}=\frac{∂}{∂y}[yz^{2}e^{xyz^{2}}]=z^{2}e^{xyz^{2}}+xyz^{4}e^{xyz^{2}}$
Differentiate $f_{xy}(x,y,z)$ with respect to $z$ keeping $x$ and $y$ constant .
$f_{xyz}=\frac{∂}{∂z}[z^{2}e^{xyz^{2}}+xyz^{4}e^{xyz^{2}}]=2ze^{xyz^{2}}+6xyz^{3}e^{xyz^{2}}+2x^{2}y^{2}z^{5}e^{xyz^{2}}$
Hence, $f_{xyz}=2ze^{xyz^{2}}(1+3xyz^{2}+x^{2}y^{2}z^{4})$
