# How do you find the first and second derivative of ln(x^(1/2))?

Nov 28, 2016

For the first derivative you can use the chain rule:

#### Explanation:

$\frac{\mathrm{dl} n \left({x}^{\frac{1}{2}}\right)}{\mathrm{dx}} = \frac{\mathrm{dl} n \left({x}^{\frac{1}{2}}\right)}{d \left({x}^{\frac{1}{2}}\right)} \cdot \frac{{\mathrm{dx}}^{\frac{1}{2}}}{\mathrm{dx}} = \frac{1}{x} ^ \left(\frac{1}{2}\right) \cdot \frac{1}{2} {x}^{- \frac{1}{2}} = \frac{1}{2 x}$

but you can also observe that:

$\ln \left({x}^{\frac{1}{2}}\right) = \frac{1}{2} \ln x$

Either way the second derivative is:

$\frac{{d}^{\left(2\right)} \ln \left({x}^{\frac{1}{2}}\right)}{{d}^{2} x} = \frac{d \left(\frac{1}{2 x}\right)}{\mathrm{dx}} = - \frac{1}{2 {x}^{2}}$