The derivative of a function $f: X \to \mathbb{R}$ for $X$ a Differentiable manifold is defined as an element of the Cotangent space, which in coordinates corresponds to the Gradient of the function. In the case of $X$ being $\mathbb{R}^n$, this corresponds to the usual derivative in Multivariable calculus.

Other derivative-like notions in Differential geometry

Differential (Jacobian, for maps between manifolds)

Vector fields. Derivative of a function (to $\mathbb{R}$) on a manifold, along a flow on the manifold, resulting on a new function on the manifold (to $\mathbb{R}$). This is basically a field of rank-1 differentials (Jacobians which are just vectors).

Lie derivative

Exterior derivative