In mathematics, the **quasi-derivative** is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative.

Let *f* : *A* → *F* be a continuous function from an open set *A* in a Banach space *E* to another Banach space *F*. Then the **quasi-derivative** of *f* at *x*_{0} ∈ *A* is a linear transformation *u* : *E* → *F* with the following property: for every continuous function *g* : [0,1] → *A* with *g*(0)=*x*_{0} such that *g*′(0) ∈ *E* exists,

If such a linear map *u* exists, then *f* is said to be *quasi-differentiable* at *x*_{0}.

Continuity of *u* need not be assumed, but it follows instead from the definition of the quasi-derivative. If *f* is Fréchet differentiable at *x*_{0}, then by the chain rule, *f* is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at *x*_{0}. The converse is true provided *E* is finite-dimensional. Finally, if *f* is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.