Difference between revisions of "Derivative rules via approximations"
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Revision as of 12:22, 21 November 2021
The fact that the derivative [math]\displaystyle{ f'(a) }[/math] exists can be written as [math]\displaystyle{ f(x) \approx f(a) + f'(a) (x-a) }[/math] for all [math]\displaystyle{ x }[/math] that are near [math]\displaystyle{ a }[/math]. This is known as linear approximation (or linearization) of f at the number a. Various rules for differentiation can be derived using linear approximation.
Constant multiple rule. Let [math]\displaystyle{ h(x) = k f(x) }[/math], where [math]\displaystyle{ k }[/math] is a constant. Then for all [math]\displaystyle{ x }[/math] near [math]\displaystyle{ a }[/math] we have [math]\displaystyle{ g(x) = kf(x) \approx k \left[ f(a) + f'(a) (x-a) \right] = kf(a) + kf'(a) (x-a). }[/math] Since [math]\displaystyle{ g(a) = kf(a) }[/math], we obtain [math]\displaystyle{ g(x)\approx g(a) + {\color{red} kf'(a)} (x-a) }[/math]. Comparing it with [math]\displaystyle{ g(x)\approx g(a) + {\color{red} g'(a)} (x-a) }[/math] we conclude that [math]\displaystyle{ g'(a) = k f'(a) }[/math], or (replacing [math]\displaystyle{ a }[/math] with an arbitrary variable):
[math]\displaystyle{ \boxed{(kf(x))' = kf'(x)} }[/math]