How do you use the definition of a derivative to find the derivative of #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#