Answer
$f_x=\frac{yz}{\sqrt (1-x^2y^2z^2)}$
$f_y=\frac{xz}{\sqrt (1-x^2y^2z^2)}$
$f_z=\frac{xy}{\sqrt (1-x^2y^2z^2)}$
Work Step by Step
Take the first partial derivatives of the given function. When taking a partial derivative with respect to x, treat y and z as constants, with respect to y, treat x and z as constants, and with respect to z, treat x and y as constants:
$f_x=\frac{1}{\sqrt (1-(xyz)^2)}\times yz=\frac{yz}{\sqrt (1-x^2y^2z^2)}$
$f_y=\frac{1}{\sqrt (1-(xyz)^2)}\times xz=\frac{xz}{\sqrt (1-x^2y^2z^2)}$
$f_z=\frac{1}{\sqrt (1-(xyz)^2)}\times xy=\frac{xy}{\sqrt (1-x^2y^2z^2)}$
