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Chapter 3: Problem 19

Find the following derivatives. $$\frac{d}{d x}(\ln (\ln x))$$

Short Answer

Expert verified
Question: Find the first derivative of \(f(x) = \ln(\ln(x))\). Answer: The first derivative of \(f(x) = \ln(\ln(x))\) is \(f'(x) = \frac{1}{x\ln(x)}\).

Step by step solution

01

Chain Rule

Get familiar with the chain rule: If you have a function \(y = g(f(x))\), then the derivative of \(y\) with respect to x is: \(\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\). In our case, the outer function will be \(g(u) = \ln(u)\), and inner function is \(f(x)=\ln(x)\).
02

Define Outer Function '

Define the outer function \(g(u)=\ln(u)\), so we can rewrite \(y = \ln(\ln(x))\), as \(y = g(f(x))\), hence apply the chain rule.
03

Differentiate Outer Function

Differentiate the outer function with respect to u: \(\frac{dg}{du}=\frac{1}{u}\)
04

Replace u

Replace \(u\) with the inner function, \(f(x) = \ln(x)\): \(\frac{dg}{du}=\frac{1}{\ln(x)}\)
05

Differentiate Inner Function

Differentiate the inner function with respect to x: \(\frac{df}{dx} = \frac{1}{x}\)
06

Apply Chain Rule

Using the chain rule \(\frac{dy}{dx}=g'(f(x))\cdot f'(x)\), we get \(\frac{dy}{dx} = \frac{1}{\ln(x)}\cdot\frac{1}{x}\)
07

Simplify the Result

Simplify the final expression: \[\frac{dy}{dx}=\frac{1}{x\ln(x)}\] The first derivative of \(\ln(\ln(x))\) is \(\frac{1}{x\ln(x)}\).

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Most popular questions from this chapter

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