How to differentiate ln2x?

Question

Ok, so im pretty confused, even though this is really simple. The question in the book is loge2x, and it says that to differentiate you use f'(x)/f(x)

Well then the answer would be 2/x right?

But according to the book the answer is 1/x. Its annoying because i have no idea. Please show me how this works. Thanks :)

Answer 1

You have to use the chain rule here.

f(x) = ln(2x) let g(x) = 2x

f'(x) = 1/(2x) g'(x)

g'(x) = 2

Putting it all together:

f'(x) = (1/(2x))(2) = 2/(2x) = 1/x

Answer 2

consider that ln [2x] = ln 2 + ln x...or use the chain rule properly....and it likely said that ln f(x) has derivative f '(x) / f(x)

Answer 3

f'(x)=2

f(x)=2x

2/2x=.....