If I take the derivative of a function in $L^ {\infty}$ (that is, the function is bounded by a number) in any direction, in which space the derivative is defined?
Are there some properties for ensuring that the derivative is also a $L ^ {\infty}$ function?
The derivative of an $L^\infty$ function belongs to $W^{-1,\infty}$, a Sobolev space of negative order. This is a pretty abstract space, which is best understood as the space of distributional derivatives of bounded functions... that is, a tautology. The elements of $W^{-1,\infty}$ are linear functionals on $W^{1,\infty}_0$; indeed, if $g\in L^\infty$ then $g'$ can be thought of as a linear functional
$$
f\mapsto -\int f' g,\quad f\in W^{1,\infty}_0
$$
If $f\in W^{1,\infty}$, then the derivative of $f$ is in $L^\infty$. The property $f\in W^{1,\infty}$ is closely related to Lipschitz continuity but there are subtleties: see Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class In any case, Lipschitz continuity implies having bounded derivative.
