# How do you sketch the graph y=ln(1/x) using the first and second derivatives?

Jan 26, 2017

$f \left(x\right) = \ln \left(\frac{1}{x}\right)$ is monotone and strictly decreasing in its domain and therefore has no local extrema, and it is concave up everywhere.

#### Explanation:

Using the properties of logarithms we should see that:

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

If we want to go through the whole sketcing process as an an exercise, we start by noting that $y = \ln \left(\frac{1}{x}\right)$ is defined and continuous for $x \in \left(0 , + \infty\right)$ and we analyze the limits at the boundaries of the domain:

${\lim}_{x \to {0}^{+}} \ln \left(\frac{1}{x}\right) = {\lim}_{y \to + \infty} \ln y = + \infty$

${\lim}_{x \to + \infty} \ln \left(\frac{1}{x}\right) = {\lim}_{y \to {0}^{+}} \ln y = - \infty$

Then we calculate the first and second derivatives:

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

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

We can see that $f \left(x\right)$ is monotone and strictly decreasing in its domain and therefore has no local extrema, and that it is concave up everywhere.

graph{ln(1/x) [-10, 10, -5, 5]}