First, this isn't a "differential equation" nor are you asked to "evaluate" anything. You are asked to find gradient of the function $\displaystyle f(x,y,z)= \frac{1}{x^2+ y^2+ z^2}= (x^2+ y^2+ z^2)^{-1}$
The derivatives are fairly straight forward. By the chain rule,
$\displaystyle \frac{\partial f}{\partial x}= -1(x^2+ y^2+ z^2)^{-2}\frac{\partial (x^2+ y^2+ z^2)}{\partial x}= \frac{-2x}{(x^2+ y^2+ z^2)^2}$
Can you do the other two partial derivatives?