Question

What is the derivative of `ln(sqrtx)`?

Answer 1

As an alternative, use `lnsqrtx = ln(x^(1/2)) = 1/2lnx`

Explanation:

So,

`d/dx(lnsqrtx) = d/dx(1/2lnx) = 1/2*1/x = 1/(2x)`

Answer 2

More detailed explanation!

`(dy/dx)=1/(2x)`

Explanation:

Pre amble:
Suppose you have`ln(z)=s`. This may be rewritten as `z=e^s`.
You also need to know that if we have `d/(ds)(e^s)` then the solution is `d/(ds)(e^s) = e^s`

Back to the question:
`y=ln(sqrt(x)) = ln(x^(1/2))`

So `x^(1/2) = e^y` or alternatively squaring both sides gives:

`x=(e^y)^2` ...................................(1)

The differential of this is:
`(dx)/(dy) = 2(e^y)` ...............................(2)

But we need `(dy/dx)` this can be achieved by inverting (2) giving

`(dy)/(dx) = [2(e^y)]^(-1) = 1/(2(e^y))`................(3)

But from (1) we have `e^y = x` so by substation in (3) we have:

`(dy/dx)=1/(2x)`