240CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATIONThis result ensures that functions defined by algebraic or analyticexpressions such as polynomials (in two or three variables) or combinationsof trigonometric, exponential, logarithmic functions and roots are generallydifferentiable, since by the formal rules of differentiation the partials areagain of this type, and hence are continuous wherever they are defined; theonly difficulties arise in cases where differentiation introduces adenominator which becomes zero at the point in question.The Gradient and Directional DerivativesRecall from§3.2that a linear function can be viewed in three differentways: as a homogeneouspolynomialof degree one, as multiplication of thecoordinate matrix by itsmatrix representative, and as thedot productofthe input with a fixed vector. We have seen that whenf(−→x) isdifferentiable at−→x=−→x0, then the coefficients of the differentiald−→x0f(−→v),
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