# How do you use the first and second derivatives to sketch F(x) = ln(x) / sqrt(x)?

$F ' \left(x\right) = \frac{2 - \ln \left(x\right)}{2 {x}^{\frac{3}{2}}}$
$F ' \left(x\right) = \frac{\frac{1}{x} \cdot \sqrt{x} - \ln \left(x\right) \cdot \frac{1}{2} \cdot {x}^{- \frac{1}{2}}}{x}$
Multiplying numerator and denominator by $\sqrt{x}$
$F ' \left(x\right) = \frac{x \cdot \frac{1}{x} - \ln \frac{x}{2}}{{x}^{\frac{3}{2}}}$
And doing the same with $2$
$F ' \left(x\right) = \frac{2 - \ln \left(x\right)}{2 {x}^{\frac{3}{2}}}$