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Derivative of Exponential and Logarithmic Functions

  1. Jan 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the equation of the tangent line to the graph of y = lnx2 at the point (2, ln4)

    2. Relevant equations



    3. The attempt at a solution

    y' = (lnx2)'
    y' = 2lnx
    y' = (2)(lnx)' + (lnx)(2)'
    y' = 2/x

    Mtan/(2,ln4) = 2/2 = 1

    Equation of tan line:
    y - ln4 = 1(x-2)
    y = x - 2 + ln4
     
  2. jcsd
  3. Jan 20, 2013 #2

    Dick

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    You forgot a prime in the second line, but that looks fine.
     
  4. Jan 20, 2013 #3
    I was looking at the answer sheet I was given along with the exercise and it says the answer should be y = x - 2 + ln

    Is it a typo ? Or I am wrong?
     
  5. Jan 20, 2013 #4
    Thanks for replying! :)
     
  6. Jan 20, 2013 #5

    Dick

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    If that's what the answer really says, then they omitted the argument of the log. Sure it's a typo.
     
  7. Jan 21, 2013 #6

    SteamKing

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    Hint: the derivative of a constant times a function is the constant times the derivative of the function:

    y = c * f(x)
    y' = c * f'(x)

    The use of the product rule in these instances is not necessary.
     
  8. Jan 21, 2013 #7

    HallsofIvy

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    You have [itex]y= ln x^2[/itex]. Is that [itex]y= ln(x^2)[/itex] or [itex]y= (ln(x))^2[/itex]? If it is the first, as you seem to have assumed, then it is easier to write it as [itex]y= 2 ln(x)[/itex] so that [itex]y'= 2/x[/itex] immediately.
     
  9. Jan 21, 2013 #8
    Oh thanks!
    The observations helped a lot. I was taking much longer writing down the whole process to find the derivatives. These tips are saving me a lot of time. THANK YOU SO MUCH!
     
    Last edited: Jan 21, 2013
  10. Jan 21, 2013 #9

    Mark44

    Staff: Mentor

    I recommend that students NEVER use the product rule when one factor of a product is a constant, as in the problem above. What SteamKing is describing is often called the constant multiple rule, which says d/dx( k *f(x)) = k * d/dx(f(x)).

    Also, you should NEVER use the quotient rule if the denominator is a constant, such as x3/6. Write this as (1/6) * x3 and use the constant multiple rule.

    The reason for these recommendations is that the product rule is more complicated than the constant multiple rule, so you are more likely to make mistakes. And the quotient rule is even more complicated, increasing your chances of making an error.
     
  11. Jan 21, 2013 #10
    I just solved one problem using this technique. Oh I am loving these tips :) they save time! :)
     
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