# Chapter 14 - Section 14.3 - Partial Derivatives - 14.3 Exercise: 65

$f_{xyz}=2ze^{xyz^{2}}(1+3xyz^{2}+x^{2}y^{2}z^{4})$

#### 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, $f_{xyz}=2ze^{xyz^{2}}(1+3xyz^{2}+x^{2}y^{2}z^{4})$

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