A directional derivative, or Gâteaux derivative, is a partial derivative of a function on a manifold along the direction given by a tangent vector.
Let $F$ and $G$ be locally convex topological vector spaces, $U \subseteq F$ an open subspace and $P\colon U \to G$ a continuous map. The derivative of $P$ at the point $f \in U$ in the direction $h \in F$ is the limit
If the limit exists for every $f \in U$ and every $h \in F$ then one can define a map
If the limit exists and $D P$ is continuous (jointly in both variables), we say that $P$ is continuously differentiable or $C^1$.
A simple but nontrivial example is the operator
given by
with the derivative
In the context of a Fréchet space, it may be that the directional derivative in every direction exists but the Fréchet derivative does not; however the existence of Fréchet derivative implies the existence of directional derivatives in all directions.
The notion of directional derivatives extends to smooth manifolds (including infinite-dimensional ones based on Fréchet spaces) using local coordinates; the differentiability does not depend on the choice of a local chart. In this case we have (if everything is defined)
where $T(U)$ is the tangent space of $U$ (an open subspace of $T(F)$.
Wikipedia (English): Gâteaux derivative
R. Gâteaux, Sur les fonctionnelles continues et les fonctionnelles analytiques, C.R. Acad. Sci. Paris Sér. I Math. 157 (1913) pp. 325–327; Fonctions d’une infinités des variables indépendantes, Bull. Soc. Math. France 47 (1919) 70–96, numdam; Sur diverses questions du calcul fonctionnel, Bulletin de la Société Mathématique de France tome 50 (1922) 1–37, numdam