How do you find the derivative of #(x+1)/(x-1)# using the limit definition?

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Answer 1 · 2016-08-04T17:42:41

#f'(x) = -2/(x-1)^2#

#f'(x) = lim_(hrarr0)(f(x+h)-f(x))/h#

#=lim_(hrarr0)((x+h+1)/(x+h-1) - (x+1)/(x-1))/h#

#=lim_(hrarr0) 1/h(((x+h+1)(x-1) - (x+h-1)(x+1))/((x+h-1)(x-1)))#

#=lim_(hrarr0) 1/h((x^2+xh-h-1 - x^2 -xh - h + 1)/((x+h-1)(x-1)))#

#=lim_(hrarr0) 1/h((-2h)/((x+h-1)(x-1)))#

#=lim_(hrarr0)(-2)/((x+h-1)(x-1)) = (-2)/((x-1)(x-1)) = -2/(x-1)^2#