If& f ( x ) has a derivative at~ x = a, then _ x → a (xf(a) -...

$\lim_{x \to a}\dfrac{xf\left(a\right) - af\left(x\right)}{x - a}$

Add and subtract~$af ( a )$~

$= \lim_{x \to a}\dfrac{xf\left(a\right) - af\left(a\right) + af\left(a\right) - af\left(x\right)}{x - a}$

$= \lim_{x \to a}\dfrac{f\left(a\right)\left(x - a\right) - a\left[f\left(x\right) - f\left(a\right)\right]}{x - a}$

Separate and solve

$= \lim_{x \to a}\dfrac{f\left(a\right)\left(x - a\right)}{x - a} - \lim_{x \to a}\dfrac{a\left[f\left(x\right) - f\left(a\right)\right]}{x - a}$

As~$f ' ( a ) = \lim_{x \to a}\dfrac{f ( x ) - f ( a )}{x - a}$~(derivative of~$f ( x )$~at~$x = a$~exist)

= f \left(a\right) - af′ \left(a\right)$</p>$