Answer
$f_x=e^{-(x^2+y^2+z^2)}\times-2x$
$f_y=e^{-(x^2+y^2+z^2)}\times-2y$
$f_z=e^{-(x^2+y^2+z^2)}\times-2z$
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:
$f_x=e^{-(x^2+y^2+z^2)}\times-2x$
$f_y=e^{-(x^2+y^2+z^2)}\times-2y$
$f_z=e^{-(x^2+y^2+z^2)}\times-2z$
