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Derivation using Natural Logs

  1. Oct 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Let y=sin(x)^x. Find dy/dx


    My teacher said to use Natural long to solve this.

    she got,
    step 1:ln(y)=ln(sin(x)^x)=x ln(sin(x))

    step 2:y(dy/dx)=1(ln(sin(x)+x(cos(x)/sin(x)

    Final answer:dy/dx=((sin(x))^x)[ln(sin(x))+x(cos x/sin x)

    I don't understand how she derived ln sin(x) to be cos(x)/sin(x)....
     
  2. jcsd
  3. Oct 3, 2012 #2

    SammyS

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    Hello gyza502. Welcome to PF !

    Use the derivative of the natural log, along with the chain rule.
     
  4. Oct 3, 2012 #3

    HallsofIvy

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    Let u= sin(x) so that y= ln(u)

    By the chain rule, dy/dx= (dy/du)(du/dx).

    What is dy/du? What is du/dx? Be sure to express every thing in terms of x, not u.
     
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