Differentiation rules

  • Adi Ben-Israel
  • Robert Gilbert
Part of the Texts and Monographs in Symbolic Computation book series (TEXTSMONOGR)


If f is differentiable, its derivative f′ can be computed using the limit (4.11),
$$f'\left( x \right) = \mathop {\lim }\limits_{\xi \to x} {\rm{ }}{{f\left( \xi \right) - f\left( x \right)} \over {\xi - x}},$$
which is often difficult. However, sometimes f has a special structure that allows differentiating it without evaluating the limit (4.11). For example, if u and v are differentiable functions, and if f is their product f = uv, then the derivative f′ can be easily computed from the derivatives u′ and v′ . This situation is covered by a differentiation rule called the product rule (Theorem 5.1). Other rules given in this chapter are the quotient rule (Theorem 5.5) and the chain rule (Theorem 5.11).


Inverse Function Product Rule Chain Rule Differentiation Rule Implicit Relation 
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Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Adi Ben-Israel
    • 1
  • Robert Gilbert
    • 2
  1. 1.Rutgers Center for Operations Research and Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Department of Mathematical Science and Computer and Informational SciencesUniversity of DelawareNewarkUSA

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