The Derivative

  • Houshang H. Sohrab


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
$$ ( * ) \frac{{f(x) - f(x_0 )}} {{x - x_0 }} = \frac{{ax + b - (ax_0 + b)}} {{x - x_0 }} = a. $$


Convex Function Open Interval Chain Rule Leibniz Rule Schwarzian Derivative 
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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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