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The Derivative

  • Houshang H. Sohrab

Abstract

The derivative is one of the two fundamental concepts introduced in calculus. The other one is, of course, the (Riemann) integral. For a real-valued function of a real variable, the derivative may be interpreted as an extension of the notion of slope defined for (nonvertical) straight lines. Recall that a (nonvertical) straight line is the graph of an affine function xax + b, where a, b are real constants and a is the slope of the line. Now, if f(x) := ax + bx ∈ ℝ, then, for any x, x0 ∈ ℝ, xx0, we have
$$ ( * ) \frac{{f(x) - f(x_0 )}} {{x - x_0 }} = \frac{{ax + b - (ax_0 + b)}} {{x - x_0 }} = a. $$

Keywords

Convex Function Open Interval Chain Rule Leibniz Rule Schwarzian Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Houshang H. Sohrab
    • 1
  1. 1.Mathematics DepartmentTowson UniversityTowson

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