The derivative of ln(x) is 1/ln(x) right?

Answer 1

No.. ln(|x|) + C is the integral of 1/x dx so the derivative of lnx is 1/x
You may be thinking of how the power rule for integration doesn't work for f(x) = x^(-1), this is the usual motovation for introducing the idea of ln x.
You may also be thinking of how integral (du/u) = ln|u| + c
As a teacher, I don't like when we have to start talking about integrals toward the end of the first semester calculus. It scrambles up the student's minds so that people tend to ask questions like this. I like it better when we save integrals till after the semester break, in calculus II, that lets the previous semester's work in derivatives gel a bit in between time.

Answer 2

No.
d(ln x)/dx = 1/x.

Let's derive the answer.

Let

y = ln x

Exponentiating
e^y = e^(ln x) = x

Differentiating
(e^y)(dy/dx) = 1
dy/dx = 1/e^y

But e^y = x and y = ln x.
So, substituting we have:

d(ln x)/dx = 1/x

Answer 3

the short answer is no.
The definition of ln(X) comes from the problem of integrating the function 1/X

Answer 4

ln(x)' = 1/x

Answer 5

No.
d lnx/dx = 1/x

Answer 6

Yes.