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

Nov 12, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x}$

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

#### Explanation:

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

We can use the law of logarithms to write:

$y = \ln x - \ln 2$

Differentiating wrt $x$ we have;

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x}$

We can rewrite this as:

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

Differentiating again wrt $x$ we have;

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

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