The Derivative

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

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