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A Simple Derivative Problem

  1. Aug 15, 2014 #1

    S.R

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    1. The problem statement, all variables and given/known data

    Assume the notation log(a, x) implies log base a of x, where a is a constant (since I don't know LaTeX).

    PROBLEM:
    If y = [log(a, x^2)]^2, determine y'.


    2. Relevant equations

    Chain Rule and Logarithmic Differentiation


    3. The attempt at a solution

    y' = 2(log(a, x^2)) * (1/[(x^2)lna]) * (2x) = (8log(a,x))/(xlna)

    Is this the correct approach and solution?
     
    Last edited: Aug 15, 2014
  2. jcsd
  3. Aug 15, 2014 #2
    Your approach and answer are both correct. If I may offer an alternate approach, try using the log law ##\log_ax^2=2\log_a x## at the beginning and see how that changes the rest of the problem. It's my experience that making good use of log laws at the beginning of some calculus problems makes them a little more manageable.
     
  4. Aug 15, 2014 #3

    S.R

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    Thanks for the response. I noticed the implementation of log laws in WolframAlpha's solution where log(a,x^2) was rewritten ln(x^2)/lna.
     
  5. Aug 21, 2014 #4

    rjr

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    gopher_p is correct: using the log law loga x2 = 2loga x does make for easier computation.

    however, if you set u = loga x2 then your equation would become y = u2

    so then, y'(x) = du/dx * 2u

    the only "tricky" part is finding du/dx, but as S.R mentioned, you can just use the change of base formula for the logarithm...which makes it much easier to find du/dx...

    u = loga x2

    = (log10 x2)/(log10 a)

    = (log x2)/(log a)

    ...then just differentiate with respect to x to find du/dx

    so yes, this is definitely the correct approach to the solution!
     
  6. Aug 22, 2014 #5

    HallsofIvy

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    Staff Emeritus
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    Of course, it is much simpler to first use the fact that [tex]log_a(x^2)= 2log_a(x)[/tex] and then differentiate.
     
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