Answer
$ \lt -\dfrac{26}{27},\dfrac{23}{54}, \dfrac{-23}{54} \gt $
Work Step by Step
Our aim is to take the first partial derivative of the given function $f(x,y,z)$ with respect to $x$, by treating $y$ and $z$ as a constant, and vice versa:
$f_x=-x \times (x^2+y^2+z^2)^{-3/2}+\dfrac{1}{x} \\ f_y= -y \times (x^2+y^2+z^2)^{-3/2}+\dfrac{1}{y} \\f_z=-z \times (x^2+y^2+z^2)^{-3/2}+\dfrac{1}{z}$
The gradient vector equation is: $\nabla f = \lt f_x,f_y,f_z \gt = \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 $
Now, $\nabla f (1,1,1) = \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 =\lt -\dfrac{26}{27},\dfrac{23}{54}, \dfrac{-23}{54} \gt $
