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Math Help - Derivative of 1/X and root X

  1. #1
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    Derivative of 1/X and root X

    can someone do these step by step please..im new at calculus and dont have my textbook with me

    so far i have (A = delta)

    root(x+Ax) - rootx / (x+Ax) - x and ive tried many different methods but i mess up

    and 1/x+Ax - 1/x / (x+Ax) -x

    i tried doing it but i keep getting stuck..and please dont use laws or anything..just algebra cause i just started
    thanks
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  2. #2
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    Quote Originally Posted by stones44 View Post
    can someone do these step by step please..im new at calculus and dont have my textbook with me

    so far i have (A = delta)

    root(x+Ax) - rootx / (x+Ax) - x and ive tried many different methods but i mess up

    and 1/x+Ax - 1/x / (x+Ax) -x

    i tried doing it but i keep getting stuck..and please dont use laws or anything..just algebra cause i just started
    thanks
    Let f(x) = \sqrt{x}.

    f(x + \delta) = \sqrt{x + \delta}

    f(x + \delta) - f(x) = \sqrt{x + \delta} - \sqrt{x}


    \frac{f(x+\delta) - f(x)}{\delta} = \frac{\sqrt{x + \delta} - \sqrt{x}}{\delta} = \frac{(\sqrt{x + \delta} - \sqrt{x})(\sqrt{x + \delta} + \sqrt{x})}{\delta (\sqrt{x + \delta} + \sqrt{x})}

     = \frac{\delta}{\delta (\sqrt{x + \delta} + \sqrt{x})} = \frac{1}{\sqrt{x + \delta} + \sqrt{x}}.


    I'm sure you can do the final step and find \lim_{\delta \rightarrow 0} \frac{f(x+\delta) - f(x)}{\delta} .
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  3. #3
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    im sorry im still kind of confused...you just put in part of that equation? the (\sqrt{x + \delta} + \sqrt{x}) part and if so, why? its seems kind of random to add that whole part


    and what about the 1/x
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  4. #4
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    Quote Originally Posted by stones44 View Post
    im sorry im still kind of confused...you just put in part of that equation? the (\sqrt{x + \delta} + \sqrt{x}) part and if so, why? its seems kind of random to add that whole part
    He multiplied 1 using the conjugate of \sqrt{x + \delta} - \sqrt{x}. This enables us to cancel the \delta in the denominator and then we can evaluate the limit.

    Follow the same procedure as Mr Fantastic's post for \frac{1}{x}. You will need to combine fractions.
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  5. #5
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    ok i got it down to
    ( 1 / (x^2 + (Dx)^2) ) - ( 1 / x^2 ) / Dx( (1/x+Dx) + 1/x )

    how to i combine the numerator
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  6. #6
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    Quote Originally Posted by stones44 View Post
    can someone do these step by step please..im new at calculus and dont have my textbook with me

    so far i have (A = delta)

    root(x+Ax) - rootx / (x+Ax) - x and ive tried many different methods but i mess up

    and 1/x+Ax - 1/x / (x+Ax) -x

    i tried doing it but i keep getting stuck..and please dont use laws or anything..just algebra cause i just started
    thanks
    Let f(x) =  \frac{1}{x}

    f(x+ \Delta x) = \frac {1}{x+ \Delta x}

    f(x+ \Delta x) -f(x) = \frac {1}{x+ \Delta x}- \frac{1}{x}

    = \frac {x-(x+ \Delta x)}{x(x+ \Delta x)}

    = \frac {- \Delta x}{x(x+ \Delta x)}

    \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}=\lim_{\Delta x \to 0} \frac {- \Delta x}{x \Delta x(x+ \Delta x)}

    =\lim_{\Delta x \to 0} \frac {- 1}{x (x+ \Delta x)}

    = \frac{-1}{x^2}
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