find second derivative of f(x) = ln(sqrt(x))

[katie9426]

I have to find the second deriviative of f(x)=ln(√x). This is what I have done for the first derivative, but I got stuck before I could get to the second derivative!! Thanks for any help you can give!

f(x)=ln(√x)
=ln(x^(1/2))
=1/x^(1/2) * (1/2)x^(-1/2) [/tex]

 

[pka]

Make it easy on yourself:
\(\displaystyle \L \ln \left( {\sqrt x } \right) = \left( {1/2} \right)\ln (x)\)

 

[katie9426]

Thanks! So I just take the second derivative of (1/2) ln(x)? The first derivative would be (1/2) * 1/x or x/2? so the second derivative would be 2/x^2?

 

[soroban]

Hello, Katie!

Be careful . . . We have: \(\displaystyle \:y\:=\:\frac{1}{2}\ln(x)\)

Then: \(\displaystyle \:\frac{dy}{dx}\:=\:\frac{1}{2}\cdot\frac{1}{x} \:=\:\frac{1}{2}x^{-1}\)

And: \(\displaystyle \:\frac{d^2y}{dx^2} \:=\:-\frac{1}{2}x^{-2} \:=\:-\frac{1}{2x^2}\)