Variation in the spectra is interpreted as compositional variation resulting from the partial differentiation of the parent body.

A partial or full differentiation between v and u, and between j and i.

Then the marginal rate of substitution can be computed via partial differentiation, as follows.

For a scalar field , covariant differentiation is simply partial differentiation:

We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that f has a continuous second derivative.

In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation .

This is especially relevant for partial differentiation.

The comma in the last part above implies the partial differentiation with respect to .

By partial differentiation of the above Lagrangian, we find:

Pallas is believed to have undergone at least some degree of thermal alteration and partial differentiation, which suggests that it was a protoplanet.