Gradient

The gradient ${\nabla }_{x}f(x)$ encodes the direction of the steepest ascent of $f$ at point $x$. It aways lives on the domain of the function $f$.

In case of a 1D function, encodes the direction on the $x$ or $y$ axis where the function grows the most.

In case of a 2D function, it encodes the direction on the plane.

Note

Note that even if we visualize the 2D function in a 3D plane, the function is still 2D. As we visualize a 1D function on a 2D plane.

The length of the gradient vector encodes the strongness of the steepest ascent, meaning how much fast the function grows.