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