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# Defition of the derivative

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I need to find the Derivative using the

formula. Im having trouble plugging them correctly into it.

Veteran  Oct 19, 2017
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(  [ (x + Δh) + 1 ] / [ (x + Δh) - 1 ]  -  [x + 1] / [ x - 1] ) / Δh

{ get a common denominator }

(  [ (x + Δh) + 1 ]   [ x - 1]  -   [x + 1]  [ (x + Δh) - 1 ] ) /  ( [ Δh  [ (x + Δh) - 1 ]  [ x - 1] )

{Simplify}

( x^2 + x Δh + x - x -  Δh - 1  -  x^2 -  xΔh + x - x -  Δh + 1 )  / ( [ Δh  [ (x + Δh) - 1 ]  [ x - 1] )

( -2Δh)  / ( [ Δh  [ (x + Δh) - 1 ]  [ x - 1] )

{ divide out   Δh  )

-2  / (   [ (x + Δh) - 1 ]  [ x - 1] )

Let  Δh → 0   and we have that the derivative is

-2  / ( x - 1)^2

WolframAlpha verifies this :

https://www.wolframalpha.com/input/?i=derivative+%5B+x+%2B+1%5D+%2F+%5B+x+-+1%5D

CPhill  Oct 19, 2017

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