Subgradient of Entropy

Question

If $(X,\Sigma\,u)$ is a finite measurable space, define the map $$ \begin{aligned} X& \rightarrow (-\infty,\infty)\\ T(f)&\triangleq \int_{x \in X} f \log(f)\nu(dx), \end{aligned} $$ where $X\subset L^1(\nu)$ is defined by $$ X\triangleq \left\{ f \in L^1(\nu) : f>0 \nu-a.e. \& T(f)<\infty \right\} . $$ How can we compute the subgradient of $f$ on its domain.