Definition for Covariant Derivative What is simple definition of the covariant derivative that looks like the definition of the derivative of a function in calculus?

definition of the derivative of a function in calculus is:

$$\frac {df(x)}{dx}=\lim_{\Delta x\to o}\frac {f(x+\Delta x)-f(x)}{\Delta x}$$

what about Covariant derivative, what is definition of the Covariant derivative of a function in calculus?
$$\frac {{\mathcal D}f(x)}{dx}$$

Remark: ok lets say that:

$$\frac {{\mathcal D}f(x)}{dx}=\frac {df(x)} {dx} +\delta f(x)$$ where the covariant derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component . Now, how do we define a simple definition for the intrinsic covariant derivative component $\delta f(x)$ (this small addition is the result of parallel translation)
There is a very intuitive way to understand the covariant derivative (for the Levi-Civita connection) for the special case of isometrically embedded submanifolds in $\mathbb R^n$. Roughly speaking, first take the usual derivative in $\mathbb R^n$, and then project the answer onto the tangent plane of the submanifold. The image of the projection is then the covariant derivative.

In fact, this is how the concept of covariant derivative if often introduced to undergraduates in their first differential geometry course. For instance, if you consider the sphere $S^2 \subset \mathbb R^3$, then it is easy (both analytically and visually) to prove that the tangent vector field to a great circle has zero covariant derivative and therefore that great circles must be geodesics.
Sounds like it is Geodesic equation:
$$\frac {d^2x^a}{ds^2}+\Gamma^a_{bc}\frac {dx^b}{ds}\frac {dx^c}{ds}$$
i really love this equation. this $\Gamma^a_{bc}\frac {dx^b}{ds}\frac {dx^c}{ds}$ is $\delta f$