derivative and antiderivative of ln(2x)

Question

Alright, if I were to find dy/dx of ln(2x) I'd get 1/2x.2, so 2/2x and that can be simplified as 1/x.

However, if I were to integrate 1/x to attempt to infer my original function, I'd get ln(x) + c.

The other option is to use 2/2x when integrating. We can't take out 2/2 to integrate the remaining 1/x as that would lead to the exact same result as earlier.

If I take out a 2, I'm left with 1/2x, and the only way that I can think of to integrate 1/2x is to take out the 1/2. My final answer would be 2. ln(x)/2, which would not lead me to my original function.

How else can I do this?

Answer 1

Your very first line is correct. You get that the derivative is 1/x.

Note that you can write ln(2x) as lnx + ln2. Here it's clear that the derivative is 1/x

Also note that you only know an indefinite integral function to an additive constant. In the case of lnx + C, it can be written as lnx + lnC' or ln(C'x)

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