  Shape Theory, together with Shape-and-Scale Theory, comprise Relational
Theory. This consists of $N$-point models on a manifold $M$, for which some
geometrical automorphism group $G$ is regarded as meaningless and is thus
quotiented out from the $N$-point model's product space $\times_{I = 1}^N M$.
Each such model has an associated function space of preserved quantities,
solving the PDE system for zero brackets with the sums over $N$ of each of
$G$'s generators. These are smooth functions of the $N$-point geometrical
invariants. Each $(M, G)$ pair has moreover a `minimal nontrivially relational
unit' value of $N$; we now show that relationally-invariant derivatives can be
defined on these, yielding the titular notions of shape(-and-scale)
derivatives. We obtain each by Taylor-expanding a functional version of the
underlying geometrical invariant, and isolating a shape-independent derivative
factor in the nontrivial leading-order term. We do this for translational,
dilational, dilatational and projective geometries in 1-$d$, the last of which
gives a shape-theoretic rederivation of the Schwarzian derivative. We next
phrase and solve the ODEs for zero and constant values of each derivative. We
then consider translational, dilational, rotational, rotational-and-dilational,
Euclidean and equi-top-form (alias unimodular affine) cases in $\geq 2$-$d$. We
finally pose the PDEs for zero and constant values of each of our $\geq 2$-$d$
derivatives, and solve a subset of these geometrically-motivated PDEs. This
work is significant for Relational Motion and Background Independence in
Theoretical Physics, and foundational for both Flat and Differential Geometry.
