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240
CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION
This result ensures that functions deFned by algebraic or analytic
expressions such as polynomials (in two or three variables) or combinations
of trigonometric, exponential, logarithmic functions and roots are generally
di±erentiable, since by the formal rules of di±erentiation the partials are
again of this type, and hence are continuous wherever they are deFned; the
only di²culties arise in cases where di±erentiation introduces a
denominator which becomes zero at the point in question.
The Gradient and Directional Derivatives
Recall from
§
3.2
that a linear function can be viewed in three di±erent
ways: as a homogeneous
polynomial
of degree one, as multiplication of the
coordinate matrix by its
matrix representative
, and as the
dot product
of
the input with a Fxed vector. We have seen that when
f
(
−→
x
) is
di±erentiable at
−→
x
=
−→
x
0
, then the coe²cients of the di±erential
d
−→
x
0
f
(
−→
v
),
as a polynomial in the entries of

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