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Math Help - quotient rule differenciate

  1. #1
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    Exclamation quotient rule differenciate

    Please help, don't know where to begin

    b(i) use the Quotient Rule to differentiate the function

    h(x) = 1+Inx
    x (x>0).

    ii) using your answer to part (b) (i), find the general solution of the differential equation

    dy/dx=(-inx/x^2)(y^1/2) (x>0,y>0)

    Give the solution in implicit form.

    (iii) Find the particular solution of the differential equation in part (b) (ii) for which y=1 when x=1, and then give this particular solution in explicit form.

    Thank you in advance
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  2. #2
    Bar0n janvdl's Avatar
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    Quote Originally Posted by student01.06 View Post
    Please help, don't know where to begin

    b(i) use the Quotient Rule to differentiate the function

    h(x) = 1+Inx
    x (x>0).
    Unfortunately I am not able to assist you with the differential equation part, but I can help you with the quotient rule.

    The quotient rule is the following:

    <br />
\left( \frac{f}{g} \right) ^{'} (x) = \frac{f'(x)g(x) - f(x)g'(x)}{\left[ g(x) \right] ^2}


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

    h'(x) = \frac{(1+ \ln x)'(x) - (1 + \ln x)(x)'}{x^2} = \frac{(\frac{1}{x})(x) - (1 + \ln x)(1)}{x^2} = \frac{- \ln x}{x^2}
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  3. #3
    MHF Contributor Reckoner's Avatar
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    Quote Originally Posted by student01.06 View Post
    ii) using your answer to part (b) (i), find the general solution of the differential equation

    dy/dx=(-inx/x^2)(y^1/2) (x>0,y>0)

    Give the solution in implicit form.
    Separation of variables:

    \frac{dy}{dx} = \frac{\left(-\ln x\right)\sqrt y}{x^2}

    \Rightarrow\int\frac{dy}{\sqrt y} = \int\left(-\frac{\ln x}{x^2}\right)dx

    Now, do you notice something about the integrand on the right?
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