Find the derivative of ln(ln(x))

\frac{d}{dx}\left(\ln\left(\ln\left(x\right)\right)\right)

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Answer

$\frac{1}{x\ln\left(x\right)}$

Step by step solution

Problem

$\frac{d}{dx}\left(\ln\left(\ln\left(x\right)\right)\right)$
1

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{d}{dx}\left(\ln\left(x\right)\right)\frac{1}{\ln\left(x\right)}$
2

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{x}\cdot\frac{1}{\ln\left(x\right)}\cdot\frac{d}{dx}\left(x\right)$
3

The derivative of the linear function is equal to $1$

$1\left(\frac{1}{x}\right)\frac{1}{\ln\left(x\right)}$
4

Any expression multiplied by $1$ is equal to itself

$\frac{1}{x}\cdot\frac{1}{\ln\left(x\right)}$
5

Multiplying fractions

$\frac{1}{x\ln\left(x\right)}$
6

Using the power rule of logarithms

$\frac{1}{\ln\left(x^x\right)}$
7

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\frac{1}{x\ln\left(x\right)}$

Answer

$\frac{1}{x\ln\left(x\right)}$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.2 seconds

Views:

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