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

Jul 6, 2016

$\therefore \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - \frac{1}{x} ^ 2$

#### Explanation:

Let $y = \ln \left(\frac{x}{2}\right)$

We have to find $\frac{{d}^{2} y}{\mathrm{dx}} ^ 2.$

We start with $y = \ln \left(\frac{x}{2}\right)$ and use Rules of Logarithmic Fun. to see that,

$y = \ln x - \ln 2$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x} = {x}^{-} 1.$

$\therefore \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{d}{\mathrm{dx}} \left\{\frac{\mathrm{dy}}{\mathrm{dx}}\right\} \ldots . \left[D e f n .\right] = \frac{d}{\mathrm{dx}} \left({x}^{-} 1\right) = - 1 \cdot {x}^{- 1 - 1} .$

$\therefore \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - \frac{1}{x} ^ 2.$