In
mathematics, the derivative is a mapping associated to another mapping, which measures the instantaneous rate of
change at a point.
Calculus is mostly the study of the derivative and its implications. When
calculus is all dressed up in its fancy clothes, it is called
analysis and constitutes a very large branch of
mathematics.
See noaseboar's writeup below for the definition.
I might add that the nth derivative of a mapping f:
X → Y can be regarded as living in the space of symmetric n-multilinear maps X → Y, which is naturally isomorphic to Hom(Symn X, Y) where Symn X is the nth symmetric power of X. See Geometric measure theory by H. Federer for a careful treatment of the multilinear algebra of geometric calculus, which can get tricky.