Question

How do you find the first and second derivative of `ln(x^(1/2))`?

Answer 1

For the first derivative you can use the chain rule:

Explanation:

`(dln(x^(1/2)))/dx= (dln(x^(1/2)))/(d(x^(1/2)))*(dx^(1/2))/dx = 1/x^(1/2)*1/2x^(-1/2)=1/(2x)`

but you can also observe that:

`ln(x^(1/2)) = 1/2lnx`

Either way the second derivative is:

`(d^((2))ln(x^(1/2)))/(d^2x)=(d(1/(2x)))/dx=-1/(2x^2)`