The Derivative

Houshang H. Sohrab
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Houshang H. Sohrab: Towson University, Mathematics Department

Chapter 6 in Basic Real Analysis, 2003, pp 209-249 from Springer

Abstract: 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 x ↦ ax + b, where a, b are real constants and a is the slope of the line. Now, if f(x) := ax + b ∀x ∈ ℝ, then, for any x, x0 ∈ ℝ, x ≠ x0, 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 (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-8176-8232-3_6

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DOI: 10.1007/978-0-8176-8232-3_6

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