Answer
$\lt -\dfrac{26}{27},\dfrac{23}{54}, \dfrac{-23}{54} \gt $
Work Step by Step
In order to find the partial derivative, we will differentiate
with respect to $x$, by keeping $y$ as a constant to find the x-coordinate of the gradient vector, and vice versa:
$f_x=-x(x^2+y^2+z^2)^{-3/2}+\dfrac{1}{x} \\ f_y= -y(x^2+y^2+z^2)^{-3/2}+\dfrac{1}{y} \\f_z=-z(x^2+y^2+z^2)^{-3/2}+\dfrac{1}{z}$
Write the gradient vector equation.
$\nabla f = \lt f_x,f_y,f_z \gt $
$\nabla f = \lt -x(x^2+y^2+z^2)^{-3/2}+\dfrac{1}{x}, -y(x^2+y^2+z^2)^{-3/2}+\dfrac{1}{y} ,-z(x^2+y^2+z^2)^{-3/2}+\dfrac{1}{z}\gt $
Thus, $\nabla f (1,1,1) = \lt -\dfrac{26}{27},\dfrac{23}{54}, \dfrac{-23}{54} \gt $
