This alternative definition is an algebraic interpretation of the
familiar limit definition from the differential calculus.
Adjoin a further indeterminate h, and consider the polynomial
f(X+h) in A[X,h]. Expand it in powers of h:
f(X+h)=f_{0}(X)+hf_{1}(X)+h^{2}f_{2}(X)+...+h^{n}f_{n}(X)
(where n is the degree of f). Clearly f_{0}(X)=f(X+0)=f(X).
By definition, f' is the polynomial f_{1}(X). That is,
f' is the unique element of A[X] for which f(X+h) is congruent to
f(X)+hf'(X) mod h^{2} in A[X,h]. It is readily checked
that f' is an A-linear function from A[X] to A[X] that takes A to 0
and X to 1 and satisfies the product rule. The formula for the
derivative of X^{n} then follows by induction on n.

**Derivations.** More generally, let B be any ring
containing A, and consider the notion of an A-linear map D on B
that satisfies the product rule D(fg)=f*Dg+g*Df. This makes sense
even if D takes values not in B but in some B-module M.
Such a function D is called a __derivation__ (or an A-derivation)
from B to M. An example from differential geometry is the map d
from the ring B of smooth functions on a manifold to the B-module
of smooth 1-forms on the same manifold
(here A=**R**, the ring of constant functions).
The set of all A-derivations from B to M is itself a B-module,
which we may call Der_{A}(B,M). For example, if B=A[X] and M=B
then Der_{A}(B,M) consists of all maps of the form D(f)=af'
for some a in B. More generally, if B is the polynomial ring in k
variables over A then Der_{A}(B,B) consists of all B-linear
combinations of the k ``partial derivatives'' with respect to those
variables.

Let D be any A-derivation on B, and let C be its kernel.
By taking f=g=1 in the product rule we see that D(1)=0.
Thus by linearity D(a)=0 for all a in A. Hence C contains A.
Also by the product rule, C is closed under multiplication,
and is thus a subring of B, called the ``ring of constants''
for the derivation D; and D is C-linear. For instance,
if A is a field, B=A[X], and D is the usual derivative,
then the ring of constants is A if the field has characteristic zero,
and A[X^{p}] if it has characteristic p.

What about the derivative on the field F(X) of rational functions in one variable? If that is to be an F-derivation from F(X) to F(X) extending the derivative on F[X], we can determine (f/g)' for any f,g in F[X] by the product rule for f=(f/g)g. Differentiating, we find f'=(f/g)g'+(f/g)'g; solving for (f/g)' yields the ``quotient rule'' (f/g)'=(f'g-fg')/g^2. We can then check that (f/g)' does not depend on the choice of representation of a given rational function as f/g, and does yield an F-derivation on F(X). More generally, if A is a subring of B which in turn is contained in somce field F, and D is an A-derivation from B to an F-vector space V, we can extend D to the quotient field of B by defining D(f/g)=(f'g-fg')/g^2, and check that this is well-defined and yields an A-derivation on that quotient field.

**What happens in algebraic field extensions?**
Now suppose K is an algebraic extension of the field F, and
D is an F_{0}-derivation from F to some K-vector space V
(where F_{0} is some subfield of F; for instance, F might be
the transcendental extension F_{0}(X), and we could have V=K
and D = the derivative from F_{0}(X) to F_{0}(X) in V).
When can we extend D to an F_{0}-derivation on K?