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

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Answer 1 · 2016-12-11T05:04:28

There are a few formulas for the definition of a derivative that follow the same idea:

#lim_(h->0)[f(x+h)-f(x)]/h#

#lim_(x->a)[f(x)-f(a)]/(x-a)#

#lim_(Deltax->0)[f(x+Deltax)-f(x)]/(Deltax)#

We'd plug in our function into one of these limits and solve.

Since #f(x)=(x+1)/(x-1)#

Then:

#lim_(h->0)[[((x+h)+1)/((x+h)-1)]-[(x+1)/(x-1)]]/h = f'(x)#

If you expand this algebraically and simplify, you should end up with:

#lim_(h->0)-2/((x+h-1)(x-1))#

Solve the limit:

Now you can plug in #h#:

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