How do you use the definition of a derivative to find the derivative of $f (x) = \frac{1}{x - 3}$ $f(x)= 1/(x-3)$ ?

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Answer 1 · 2016-01-14T10:10:47

$f'(x):=lim_(h rarr 0)(f(x+h)-f(x))/h=-1/(x-3)^2$

$f'(x):=lim_(h rarr 0)(f(x+h)-f(x))/h$

$f(x)=1/(x-3)$

$:. f'(x)=lim_(h rarr 0)(1/((x+h)-3)-1/(x-3))/h=$

$=lim_(h rarr 0)1/h*((x-3-(x+h-3))/((x+h-3)(x-3)))=$

$=lim_(h rarr 0)1/h*((color(green)color(green)cancel(x)color(magenta)cancel(-3)color(green)cancel(-x)-hcolor(magenta)cancel(+3)))/((x+h-3)(x-3))=$

$=lim_(h rarr 0)1/color(green)cancel(h)*(-color(green)cancel(h)/((x+h-3)(x-3)))=$

$=color(red) - lim_(h rarr 0)1/((x+h-3)(x-3))=-1/(x-3)^2$

How do you use the definition of a derivative to find the derivative of f(x)= 1/(x-3)f(x)=1x−3f(x)= 1/(x-3)?