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Thread: The natural logarithmic function and differentiation?

  1. #1
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    The natural logarithmic function and differentiation?

    Can anybody show me how to find the derivative of this function

    ((-square root(x^2+1))/((x))+ln((x+square root(x^2+1))

    How can I do this one I have thought about using the quotient rule.

    Any help especially one that leads to a solution would be appreciated.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: The natural logarithmic function and differentiation?

    If the function is:
    f(x)=\frac{-\sqrt{x^2+1}}{x}+\ln\left(x+\sqrt{x^2+1}\right)

    Yes, first notice it's a sum so you can take the sum of the derivatives of each term, then use the quotientrule for the first one and also the chain rule where necessary.

    Important is that you show some work of yourself.
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  3. #3
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    Re: The natural logarithmic function and differentiation?

    Hmm so I I use the quotient rule for the first one I get

    ((x 2x/square root(x^2+1)-square root(x^2+1))/((x^2))

    And the second one I apply the chain rule?

    then for my second one I have

    ln(x)+ln square root(x^2-1)

    y=(1/x)+(1/2)(2x/x^2-1)
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: The natural logarithmic function and differentiation?

    Yes, We start with the first term:
    \frac{d}{dx}\left[\frac{-\sqrt{x^2+1}}{x}\right]=\frac{\frac{-x^2}{\sqrt{x^2+1}}+\sqrt{1+x^2}}{x^2}
    =\frac{1}{x^2 \cdot \sqrt{1+x^2}}

    For the second one:
    \frac{d}{dx}\ln(x+\sqrt{x^2+1})=\frac{\frac{d}{dx}  (x+\sqrt{x^2+1})}{x+\sqrt{x^2+1}}=...

    Notice:
    \ln(a\cdot b)=\ln(a)+\ln(b) and \ln(a+b) \neq \ln(a)+\ln(b)
    Can you continue?
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