COnfused: what is the derivative of ln(2x)?

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

    What is the derivative of ln(2x)?

    I was just thinking about this, and I got 2 answers. I am in Calc 2 right now.

    2. Relevant equations

    Derivative of ln(x) = 1/x


    3. The attempt at a solution

    Since d/dx lna = (1/a)*(derivative of a)

    Thus d/dx ln2x = (1/2x)*(2)

    BUT

    I can also do this, I think: d/dx ln2x = 2d/dx lnx = 2*1/x = 2/x

    I am CONFUSED!! lol !:)

    Please tell me which is the correct method! :)

    Thanks! :)
     
  2. jcsd
  3. both the methods are incorrect
    d/dx(log 2x)=(1/2x)d/dx(2x)
    =1/x
     
  4. This is correct. Note that ln(ax) = ln(a) + ln(x). Since ln(a) is a constant, the derivative is always 1/x, irrespective of 'a'. In geometric terms, 'a' simply moves the graph of the logarithm up or down; it does not change the shape of the graph.

    This is wrong. The natural logarithm is not linear: you cannot pull the 2 out of the ln, irrespective of the derivative. ln(2x) is not 2ln(x) any more than cos(2x) = 2cos(x). It would be a good idea to review the definition and properties of logarithms.
     
  5. Thanks Slider and Monty!! :)
     
  6. (d(ln 2x)/ dx) / (d(2x)/ dx) = 2/2x/2 = 1/2x
     
  7. HallsofIvy

    HallsofIvy 40,303
    Staff Emeritus
    Science Advisor

    100% wrong! Go back and read the previous responses to this question. The derivative is 1/x.
     
  8. ohh sorry I calculatd, derivative of ln2x wrt to 2x.
     
  9. 1/2x
     
  10. lanedance

    lanedance 3,307
    Homework Helper

    try reading the other posts... d(ln2x)/dx = 1/x
     
  11. anti derivative of 1/x or x^-1 = ln (x) natural log of x =ln x +c so the derivative of c + ln (2x)dx=1/2x +C'
     
  12. Mark44

    Staff: Mentor

    Wrong on two counts:
    1. d/dx(c) = 0 - not c'
    2. d/dx(ln(2x)) = 1/x - you are forgetting to use the chain rule.
     
  13. jambaugh

    jambaugh 1,800
    Science Advisor
    Gold Member

    I didn't see it mentioned but observe also you can apply the properties of logarithms:

    [tex] d/dx \, \ln(2x) = d/dx\, [\ln(x) + \ln(2)] = 1/x + 0[/tex]
     
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