How do you use the limit definition to find the derivative of #f(x)=(x+1)/(x-1)#?

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Answer 1 · 2018-06-11T22:54:21

# d/dx ((x+1)/(x-1)) = -2/(x-1)^2#

#(df)/dx = lim_(h->0) (f(x+h)-f(x))/h#

so:

#(df)/dx = lim_(h->0) 1/h( (x+h+1)/(x+h-1)-(x+1)/(x-1))#

#(df)/dx = lim_(h->0) 1/h( (x+h+1)(x-1)-(x+h-1)(x+1))/((x+h-1)(x-1))#

#(df)/dx = lim_(h->0) 1/h( color(blue)((x+1)(x-1))+h(x-1) -color(blue)((x-1)(x+1))-h(x+1))/((x+h-1)(x-1))#

#(df)/dx = lim_(h->0) 1/h(color(blue)(hx)-h - color(blue)(hx)-h)/((x+h-1)(x-1))#

#(df)/dx = lim_(h->0) 1/color(blue)(h)(-2color(blue)(h) )/((x+h-1)(x-1))#

#(df)/dx = lim_(h->0) (-2 )/((x+h-1)(x-1)) =-2/(x-1)^2#