Question

Differentiate \(f\) and find the domain of \(f\).

36.\(f\left( x \right) = \ln \ln \ln x\)

Short answer

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

Step-by-step solution

Write the formula of the derivatives of logarithmic functions, the chain rule, and the power rule

Derivative of logarithm function: \(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\),

The Chain Rule:If the function \(g\) is a differentiable function at \(x\) and \(f\) is a differentiable function at \(g\left( x \right)\) the composite function\(F = f \circ g\) defined by \(F\left( x \right) = f\left( {g\left( x \right)} \right)\) is a differentiable function at \(x\) and \(F'\) is given by the product shown below, \(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\).

Domain: The domain is the set of all the values for which the function is well defined.

Find the differentiation of the function

Consider the function\(f\left( x \right) = \ln \ln \ln x\). Differentiate the function w.r.t \(x\) by using the derivatives of logarithmic functions.

\(\begin{aligned}{c}\frac{d}{{dx}}\left( {f\left( x \right)} \right)&= \frac{d}{{dx}}\left( {\ln \left( {\ln \ln x} \right)} \right)\\&= \frac{1}{{\ln \ln x}}\frac{d}{{dx}}\left( {\ln \ln x} \right)\\&= \frac{1}{{\ln \ln x}} \cdot \frac{1}{{\ln x}}\frac{d}{{dx}}\left( {\ln x} \right)\\&= \frac{1}{{\ln \ln x}} \cdot \frac{1}{{\ln x}} \cdot \frac{1}{x}\end{aligned}\)

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

Find the domain of \(f\)

Consider the function\(f\left( x \right) = \ln \ln \ln x\) then the domain is:

\(\begin{aligned}{c}{\rm{Dom}}\left( f \right)&= \left\{ {x\left| {\ln \ln x > 0} \right.} \right\}\\&= \left\{ {x\left| {\ln x > 1} \right.} \right\}\\&= \left\{ {x\left| {x > e} \right.} \right\}\\&= \left( {e,\infty } \right)\end{aligned}\)

Thus,the domain of the function\(f\left( x \right) = \ln \ln \ln x\) is \(\left( {e,\infty } \right)\).