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complicated because it contains so many variables. One needs to look beyond the
notation. We are simply defining a linear map T(x,y), but one that varies as we move
from one point (a,b) to another. If the reader will bear with us, we shall soon intro-
duce a more common and more compact notation for the derivative, namely the
Jacobian matrix of partial derivatives, that will eliminate the excess of symbols.
The advantage of the current notation however is that there is absolutely no ambi-
guity about it whereas the same cannot be said about some instances of the partial
derivative notation.
Now, since we clearly do not want to compute derivatives via limits, we need some
basic facts that will aid in their computation.
4.3.4. Proposition
(1) If f is a constant map, then Df = 0 .
(2) If f is a linear transformations, then Df = f.
(3) The derivative at the point (a,b) of the map f(x,y) = xy is defined by Df(a,b)(x,y)
= bx + ay.
(4) If f and g are two differentiable functions with the same domain and range,
then f + g is differentiable and D(f + g) = Df + Dg.
Proof. The proposition follows in a straightforward manner from the definition of
the derivative and is left as an exercise (Exercise 4.3.1).
The next proposition reduces the problem of finding the derivative of a vector-
valued function to finding the derivative of real-valued functions of several variables.
Let U be an open subset of R n
and let f : U Æ R m . If f( p ) =
4.3.5. Proposition.
(f 1 ( p ),...,f m ( p )), where f i : R n
Æ R , then f is differentiable at p if and only if each f i
is and
( () =
(
( ()
( ()
)
Df
ph
Df
ph
,...,
Df m
ph
.
1
Proof.
This is again straightforward. See [Spiv65].
4.3.6. Theorem. (The Chain Rule) Let U and V be open sets in R n and R m , respec-
tively. Suppose that we have maps f : U Æ R m , g : V Æ R k and that f( U ) Õ V . If f is
differentiable at p in U and g is differentiable at q = f( p ), then the composite
g f: R n
Æ R k is differentiable at p and
(
)( ) =
()
()
Dg f
o
p
Dg
q
o
Df
p .
Proof.
See [Spiv65]. The proof is similar to the one for functions of one variable.
It is very important that one understands the chain rule for derivatives. Not only
does it lead to many differentiation formulas (starting with only a few basic ones) but
it gets used over and over again in practical problems. Lots of nice formulas would
be impossible without it. One immediate corollary is that the derivative of vector-
valued functions satisfies the usual properties of a derivative.
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