tex]...

July 2022 1 20 Report

Найти производную функции, пользуясь непосредственно определением производной.
y = [tex] \frac{4}{ (x-4)^{2} } [/tex]

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sergpiv $y^{'}(x_0)= \lim_{x \to \x x_0} \frac{y(x)-y(x_0)}{x-x_0}= \lim_{x \to \x x_0} \frac{ \frac{4}{(x-4)^2}- \frac{4}{(x_0-4)^2} }{x-x_0}=$
$=\lim_{x \to \x x_0} \frac{ \frac{4(x_0-4)^2-4(x-4)^2}{(x-4)^2(x_0-4)^2} }{x-x_0}=\lim_{x \to \x x_0} \frac{ \frac{4(x_0-4-x+4)(x_0-4+x-4)}{(x-4)^2(x_0-4)^2} }{x-x_0}=$
$=\lim_{x \to \x x_0} \frac{ \frac{4(x_0-x)(x_0+x-8)}{(x-4)^2(x_0-4)^2} }{x-x_0}=\lim_{x \to \x x_0} \frac{-4(x_0+x+8)}{(x-4)^2(x_0-4)^2}= \frac{-4(2x_0-8)}{(x_0-4)^4}=$
$= \frac{-8(x_0-4)}{(x_0-4)^4} = \frac{-8}{(x_0-4)^3}$

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