How do you find the second derivative of ln(sqrtx)?

2 Answers
May 13, 2015

In order to derivate a ln we must remember its derivation rule: dlnf(x) = (f'(x))/f(x)

Let's just rewrite sqrt(x) as x^(1/2). Like this: ln(x^(1/2))

Now, let's derivate it.

f'(x) = (1/2).x^(-1/2)

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

Now we've found f'(x), let's solve the first derivative:

(dln(x^(1/2)))/(dx) = (1/(2x^(1/2)))/(x^(1/2))

Simplifying it for exponential rules: dln(x^(1/2)) = 1/(2x)

Deriving (1/(2x)), again, to obtain your original function's second derivative:

(d²ln(x^(1/2)))/(dx²) = (-1/(2x^(2))) (or, alternatively, -1/2 x^(-2))

May 13, 2015

sqrtx = x^(1/2), so

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

So the derivative is 1/2*1/x=1/2 x^(-1)

The second derivative, then, is

-1/2x^-2 = -1/(2x^2)