Derivative rules via approximations
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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 x that are near a. This is known as linear approximation (or linearization) of f at the number a. Various rules for differentiation can be derived using linear approximation.