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Math Help - first derivative with natural log

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    first derivative with natural log

    Hi, I just spent a while doing math homework and I am really stuck on this last question. It reads find the first derivative of

    y = x^ln(x)

    can someone please help me so i can move on.. Thanks!!!
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    Quote Originally Posted by Cucc1137 View Post
    Hi, I just spent a while doing math homework and I am really stuck on this last question. It reads find the first derivative of

    y = x^ln(x)

    can someone please help me so i can move on.. Thanks!!!
    you have to do logarithmic differentiation here. so begin by logging both sides. we get,

    \ln y = \ln x^{\ln x}

    \Rightarrow \ln y = \ln x \ln x

    \Rightarrow \ln y = (\ln x)^2

    now can you continue?
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    Quote Originally Posted by Cucc1137 View Post
    Hi, I just spent a while doing math homework and I am really stuck on this last question. It reads find the first derivative of

    y = x^ln(x)

    can someone please help me so i can move on.. Thanks!!!
    \ln y = \ln x \ln x = (\ln x)^2

    Differentiate both sides with respect to x (use implicit differentiation on the left hand side):

    \frac{1}{y} \frac{dy}{dx} = 2 \, (\ln x) \, \left( \frac{1}{x}\right).

    Now make \frac{dy}{dx} the subject and substitute y = x^{\ln x}.
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    is that last step the derivative or do you take the derivative of that part?
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    Quote Originally Posted by Cucc1137 View Post
    is that last step the derivative or do you take the derivative of that part?
    I have no idea what you mean.

    I gave you that

    \frac{1}{y} {\color{red}\frac{dy}{dx}} = 2 \, (\ln x) \, \left( \frac{1}{x}\right).

    Now make the stuff in red the subject (by multiplying both sides by y).
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  6. #6
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    so is it

    dy/dx = (2)(lnx)(1/x)(x^lnx)

    can u simplify that down alot of just the basic (2lnx/x)(x^lnx) ???
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    Quote Originally Posted by Cucc1137 View Post
    so is it

    dy/dx = (2)(lnx)(1/x)(x^lnx)

    can u simplify that down alot of just the basic (2lnx/x)(x^lnx) ???
    Yes and yes.
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    so what i have at the end is right .. or can you simplify more??
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    Quote Originally Posted by Cucc1137 View Post
    so what i have at the end is right Mr F says: Yes. .. or can you simplify more?? Mr F says: No.
    If it could be simplified further I would have suggested that to you.
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  10. #10
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    o alright.. thank you
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