Chapter 3.7, Problem 24E

To determine

To calculate: The derivative and the domain of the function $f\left(x\right)=\ln \ln \ln x$ .

Answer to Problem 24E

The derivative of the function is $f\text{'}\left(x\right)=\frac{1}{x\ln \left(\ln x\right)\ln x}$ and the domain of the function is $\left(e,\infty \right)$ .

Explanation of Solution

Given information:

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

Formula used:

Thechain rule for differentiation is if f is a function of gthen $\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)=f\text{'}\left(g\left(x\right)\right)g\text{'}\left(x\right)$ .

Power rule for differentiation is $\frac{d}{dx}{x}^n=n{x}^{n-1}$ .

Product rule for differentiation is $\frac{d}{dx}\left(f\cdot g\right)=f\text{'}\left(x\right)g\left(x\right)+f\left(x\right)g\text{'}\left(x\right)$ where f and g are functions of x .

Quotient rule for differentiation is $\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g\left(x\right)f\text{'}\left(x\right)-f\left(x\right)g\text{'}\left(x\right)}{{\left[g\left(x\right)\right]}^2}$ where f and g are functions of x .

Domain of the function is the set of values of independent variable for which function is defined.

Calculation:

Consider the function $f\left(x\right)=\ln \ln \ln x$ .

Differentiate both sides with respect to x ,

  $\begin{array}{c}\frac{d}{dx}\left(f\left(x\right)\right)=\frac{d}{dx}\left(\ln \ln \ln x\right)\\ f\text{'}\left(x\right)=\frac{d}{dx}\left(\ln \ln \ln x\right)\end{array}$

Recall that chain rule for differentiation is if f is a function of gthen $\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)=f\text{'}\left(g\left(x\right)\right)g\text{'}\left(x\right)$ .

Apply it.

  $\begin{array}{c}f\text{'}\left(x\right)=\frac{d}{dx}\left(\ln \left(\ln \ln x\right)\right)\\ f\text{'}\left(x\right)=\frac{1}{\ln \ln x}\frac{d}{dx}\left(\ln \ln x\right)\\ =\frac{1}{\ln \ln x}\left(\frac{1}{\ln x}\right)\frac{d}{dx}\left(\ln x\right)\\ =\frac{1}{\ln \ln x}\left(\frac{1}{\ln x}\right)\left(\frac{1}{x}\right)\end{array}$

Simplify it further as,

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

Thus, the derivative of the function is $f\text{'}\left(x\right)=\frac{1}{x\ln \left(\ln x\right)\ln x}$ .

Consider the function, $f\left(x\right)=\ln \ln \ln x$ .

Recall that domain of the function is the set of values of independent variable for which function is defined.

The logarithmic function is defined only for positive values.

The provided expression $f\left(x\right)=\ln \ln \ln x$ is defined for positive that is,

  $\begin{array}{c}\ln \ln \ln x>0\\ \ln \ln x>e^0\\ \ln x>e^1\\ x>e\end{array}$

Write it in interval notation.

Therefore, domain is $\left(e,\infty \right)$ for function to be defined.

Hence, the domain of the function $f\left(x\right)=\ln \ln \ln x$ is $\left(e,\infty \right)$ .