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CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
This result ensures that functions defined by algebraic or analytic
expressions such as polynomials (in two or three variables) or combinations
of trigonometric, exponential, logarithmic functions and roots are generally
differentiable, since by the formal rules of differentiation the partials are
again of this type, and hence are continuous wherever they are defined; the
only difficulties arise in cases where differentiation 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 different
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 fixed vector. We have seen that when
f
(
−→
x
) is
differentiable at
−→
x
=
−→
x
0
, then the coefficients of the differential
d
−→
x
0
f
(
−→
v
),
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 Fall '08
 ALL
 Calculus, Polynomials, Derivative, Logarithmic Functions, Dot Product, representative, jf, matrix representative

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