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Summary: Differentiation with respect to a coordinate.
A numerical function is a function whose domain and range are subsets of R, the set of real numbers.
Suppose f is a numerical function. We say f is differentiable at a a is an interior point of the domain of
f and
lim
xa
f(x) - f(a)
x - a
exists. The derivative of f denoted
f
is, by definition, the set of ordered pairs (a, b) of real numbers f is differentiable at a and
b = lim
xa
f(x) - f(a)
x - a
.
Evidently, the domain of f is the set of points in the domain of f at which f is differentiable; this set could
be empty, in which case f is the empty function.
A very important fact about differentiation of numerical functions is the following.
The Chain Rule. Suppose
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