# How do you find the first and second derivative of ln(x/20)?

Jan 18, 2017

$\frac{d}{\mathrm{dx}} \ln \left(\frac{x}{20}\right) = \frac{1}{x}$

#### Explanation:

Based on the properties of logarithms:

$\ln \left(\frac{x}{20}\right) = \ln x - \ln 20$

so that:

$\frac{d}{\mathrm{dx}} \ln \left(\frac{x}{20}\right) = \frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$

Jan 18, 2017

We use the Rule $\ln \left(\frac{a}{b}\right) = \ln a - \ln b ,$ and get, $y = \ln x - \ln 20.$

$\therefore \text{ The First Derivative } y ' = \left(\ln x\right) ' - \left(\ln 20\right) ' = \frac{1}{x} - 0$

$= \frac{1}{x} = {x}^{-} 1.$

$\text{Next, since, "(x^n)'=nx^(n-1)," the Second Derivative } y ' '$

$= \left(y '\right) ' = \left({x}^{-} 1\right) ' = - 1 {x}^{- 1 - 1} = - \frac{1}{x} ^ 2$.