`f(x) = ln ln ln x` Differentiate f and find the domain of f.

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Chapter 3, 3.6 - Problem 30 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sbernabel | Student, Graduate | (Level 1) Adjunct Educator

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`f(x)=ln(ln(ln(x))) `. To find the domain of `f(x)` we need to solve the inequality `ln(ln(x))>0 `. Take the exponential on both sides, `e^(ln(ln(x)))>e^0=>ln(x)>1=>e^(ln(x))>e^1=>x>e `. The plot shows this result:

Next, we need to apply the Chain rule multiple times, keeping in mind the derivative of `ln(x)` which is `1/x `. Thus,

`d/dx f(x) = (1)/(ln(ln(x)))(ln(ln(x)))' = `

`(1)/(ln(ln(x)))*(1)/(ln(x))(ln(x))'= `

`(1)/(ln(ln(x)))*(1)/(ln(x))*1/x = (1)/(x ln(x) ln(ln(x)))`


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