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Math Help - logarithmic differentiation

  1. #1
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    logarithmic differentiation

    how do i differentiate this:

    y=sin(x^sinx)

    so far i've manage to do the following:

    ln(y)=ln(sin(x^sinx))

    but then i reach

    \frac{1}{y} \frac{dy}{dx }=?

    and im not sure how to differentiate the RHS

    Thanks!
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  2. #2
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    try the following:

    CHAIN RULE: y=sin(x^sin(x)) y'=cos(x^sin(x))*x^sin(x)' = (I); now we forget the rest and only look at x^sin(x)

    y=x^sin(x)
    ln(y)=ln(x^sin(x))=sin(x)*ln(x) (as ln(a^b)=b*ln(a))
    Now we do some differntiating
    ln(y)'=y'/y, sin(x)*ln(x)'=sin(x)'*ln(x)+sin(x)*ln(x)'=cos(x)ln (x)+sin(x)/x

    ->y'/y=cos(x)ln(x)+sin(x)/x
    Now the interesting part. We are not looking for ln(y)' but (as f(x)=y) we are looking for y', which we can find in the equation above. Hence, u solve the equation for y'
    y'=y(cos(x)ln(x)+sin(x)/x)=x'sin(x)*(cos(x)ln(x)+sin(x)/x) (II)
    Now plug (II) in in (I) and you have the solution
    (x^sin(x)*cos(x^(sin(x))*(cos(x)ln(x)+sin(x))/x


    Generally the point of logarithmic differentiation is the following:
    f(x)=y=a^x
    -> ln(y)=a*ln(x)
    -> ln(y)'=(a*ln(x))'
    -> y'/y = (a'ln(x) + a/x)
    ->y'=y*(a'ln(x) + a/x)
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  3. #3
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    i don't get the last part where u subs (II) into (I)? how do u manage to do that?
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  4. #4
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    Quote Originally Posted by Rine198 View Post
    i don't get the last part where u subs (II) into (I)? how do u manage to do that?
    You know that \displaystyle y = \sin{\left(x^{\sin{x}}\right)}, so you substitute this to get your derivative explicitly in terms of \displaystyle x.
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  5. #5
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    The easiest way to apply logarithmic differentiation is to a function in the form of f(x)=a^g(x), thus in your case x^sin(x). Henceforth you solve the solution until the only thing left is to differntiate x^sin(x), which is done in (I). sin(x^sin(x))= cos(x^sin(x))*x^sin(x)' = cos(x^sin(x)*(II))
    The way how to differntiate x^sin(x) is shown in my previous thread, whereas the solution to (x^sin(x))' equals (II).
    Thus the soultion to the whole problem is cos(x^sin(x)*x^sin(x)*(cos(x)ln(x)+(sin(x)/x))
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