Question

What is the derivative of `f(x) = ln(lnx)`?

Answer 1

`(dy)/dx=1/(xlnx)`

Explanation:

Let `y=lnu` then `(dy)/(du)=1/u`

Let `u=lnx` then `(du)/dx=1/x`

By the chain rule

`(dy)/dx=(dy)/(du)(du)/dx=1/u(1/x)=1/lnx(1/x)=1/(xlnx)`

Answer 2

The derivative is `1/(xln(x))`

Explanation:

You have a composite function `f(g(x))`, with the special case in which `f(x)=g(x)=ln(x)`

So, the chain rule states that

`(f(g(x))' = f'(g(x)) * g'(x)`,

and we need to use the fact that the derivative of `ln(x)` is `1/x`.

So, we have that `f'(g(x)) = 1/g(x)=1/ln(x)`, and `g'(x)=1/x`.

Putting everything together, we have

`d/dx ln(ln(x))= 1/ln(x) * 1/x = 1/(xln(x))`