Differentiation

Abstract

The differential quotient of a function y = f(x) at x₀ is equal to $\mathop {\lim }\limits_{\Delta x \to 0} \frac{{f(x_0 + \Delta x) - f(x_0 )}} {{\Delta x}}$ if this limit exists and is finite. The derivative function of a function y = f(x) with respect to the variable x is another function of x denoted by the symbols $y',\dot y,Dy,\frac{{dy}} {{dx}},f'(x),Df(x),or\frac{{df(x)}} {{dx}}$, and its value for every x is equal to the limit of the quotient of the increment of the function Δy and the corresponding increment Δx for Δx→ 0, if this limit exists:

$$ f'(x) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f(x + \Delta x) - f(x)}} {{\Delta x}}. $$

((6.2))