"Find the derivatives of the functions. f(x)=ln⁡|ln⁡|ln⁡x||" was answered by our experts

CMIIh

Answered question

2021-05-26

Find the derivatives of the functions. $f (x) = \ln | \ln | \ln x | |$

likvau

Skilled2021-05-27Added 75 answers

Step 1

The function we want to differentiate is

$f (x) = \ln | \ln | \ln x | |$

Step 2

Recall the generalized rule for the derivative of a natural logarithm of an absolute value

$\frac{d}{d x} [\ln | u |] = \frac{1}{u} \frac{d u}{d x}$

Step 3

Now we get

$f^{'} (x) = \frac{1}{\ln | \ln x |} \frac{d}{d x} (\ln | \ln x |)$

Step 4

In Exercise 73 , we computed

$\frac{d}{d x} (\ln | \ln x |) = \frac{1}{x \ln x}$

Step 5

Therefore $f^{'} (x) = \frac{1}{\ln | \ln x |} \cdot \frac{1}{x \ln x} = \frac{1}{x \ln x \ln | \ln x |}$

Result

$f^{'} (x) = \frac{1}{x \ln x \ln | \ln x |}$

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