Q. 4 - Calculus

Question

Sign analyses for derivatives: For each function f that follows, find the derivative $f\text{'}.$ Then determine the intervals on which the derivative $f\text{'}$  is positive and the intervals on which the derivative $f\text{'}$ is negative. Record your answers on a sign chart for $f\text{'},$ with tick-marks only at the x-values where $f\text{'},$  is zero or undefined.

$f\left(x\right)=\ln \left(\ln x\right)$

Step-by-Step Solution

Answer

The derivative of the function is defined at $x>1$ and its derivative is always positive. 

Step 1. Given Information.

The given function is $f\left(x\right)=\ln \left(\ln x\right).$

Step 2. Find the derivative.

To find the derivative of the given function, we will use the chain rule of differentiation.

So,

$$\begin{gathered} f\left(x\right)=\ln \left(\ln x\right) \\ f\text{'}\left(x\right)=\frac{1}{x\left(\ln x\right)} \end{gathered}$$

Now, let's find the critical point by putting the above equation to zero,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{1}{x\left(\ln x\right)} \\ 0=\frac{1}{x\left(\ln x\right)} \\ 0\ne 1 \end{gathered}$$

Thus, there are no critical points.

The derivative of the function is always positive.