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Math Help - derivative

  1. #1
    Senior Member euclid2's Avatar
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    derivative

    find the derivative of y=\frac{x+1}{x-1} using first principles

    so

    \frac{f(x+h)-f(x)}{h}

    <br />
\frac{(x+h)+1}{(x+h)-1} - \frac{x+1}{x-1} all over h

    can't get the right answer
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  2. #2
    MHF Contributor
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    Quote Originally Posted by euclid2 View Post
    find the derivative of y=\frac{x+1}{x-1} using first principles

    so

    \frac{f(x+h)-f(x)}{h}

    <br />
\frac{(x+h)+1}{(x+h)-1} - \frac{x+1}{x-1} all over h

    can't get the right answer
    f(x+h)-f(x)=\frac{x+1+h}{x-1+h}-\frac{x+1}{x-1}

    =\frac{(x-1)(x+1+h)-(x+1)(x-1+h)}{(x-1+h)(x-1)}

    =\frac{\color{blue}(x-1)(x+1)\color{black}+(x-1)h\color{blue}-(x+1)(x-1)\color{black}-(x+1)h}{(x-1)(x-1)+h(x-1)}

    =\frac{xh-h-xh-h}{(x-1)^2+h(x-1)}

    =\frac{-2h}{(x-1)^2+h(x-1)}

    \frac{f(x+h)-f(x)}{h}=\frac{-2}{(x-1)^2+h(x-1)}

    setting h=0, the derivative is

    f'(x)=\frac{-2}{(x-1)^2}
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