Covariant derivatives have normally been defined via Christoffel symbols $\Gamma _{jk}^i $ which have usually been introduced beforehand in one of three ways : (i) in terms of fundamental tensors (e.g., Sokolnikoff [5, p. 76]). The second and third ways employ the symbols as “coefficients of connection”: (ii) as coefficients on base vectors in a linear expression for the derivatives of the base vectors (e.g., Sedov [4, p. 6]) ; (iii) as coefficients which reflect the change along a coordinate curve of the components of a constant vector (e.g., Weyl [6, p. 114]). Thus, heretofore, Christoffel symbols are invented beforehand, knowing that they will be the key to obtaining the desired result. Here, a more natural and hence more simple development of derivatives of tensor fields is presented, based upon the explicit recognition of a relationship between the base vectors at different points and, most importantly, upon the direct application of partial derivatives. This direct application is made possible by using “multiple-point functions” to express the components of tensors at one point relative to the base vectors at another–functions which occur naturally, inasmuch as connected manifolds are a corequisite for tensor fields. The covariant derivative components then arise as the directional derivatives (along coordinate curves) with respect to the first point in the argument, with the other points held fixed. The equivalent of a Christoffel symbol arises naturally as components of the derivatives of covariant base vectors. The more simple development given here generalizes the conclusions which Sokolnikoff [5, p. 125] draws for Euclidean manifolds, to non-Euclidean manifolds-in fact, to manifolds which are not even metrized.