Question

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

Answer 1

The second derivative is

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

Note that `lnsqrtx=lnx^(1/2)=1/2*lnx`

Answer 2

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

Explanation:

We can simplify `f(x)` through the rule that

`log_a(b^c)=c*log_a(b)`

Here, we see that

`f(x)=lnsqrtx=ln(x^(1/2))=1/2ln(x)`

So, to find the derivative of this function, we must know that the derivative of `ln(x)` is `"1/"x`, so

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

To differentiate this and find the second derivative, use the power rule, recalling that `"1/"x=x^-1`.

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