If \(f(x)=\ln(\sqrt{x})\), then \(f''(x)=\)

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Rewrite \(f(x)\) into fractional exponent form.

$$ f(x)=\ln(\sqrt{x})=\ln{x^{\frac{1}{2}}} $$ $$ f(x)= \frac{1}{2}\ln{x} \tag*{\tiny log of power rule}$$

Calculating the first derivative,

$$ f'(x)= \frac{1}{2}\cdot \frac{1}{x} $$ $$ f'(x)= \frac{1}{2x} $$

Again, rewrite into fractional exponent form to calculate the second derivative,

$$ f''(x)=\frac{1}{2}x^{-1} $$ $$ =-1 \cdot \frac{1}{2} \cdot x^{-2} $$ $$ = \boxed{-\frac{1}{2x^2}}$$