Derivatives Evaluate the following derivatives. ddx(ln⁡(ln⁡x))

Question

Derivatives Evaluate the following derivatives.  ddx(ln⁡(ln⁡x))

Louis Page · Accepted Answer

Step 1 : Analysis  Given:  ddx(ln⁡(ln⁡x))  Step 2 : Solution  Using d(ln⁡x)dx=1x and chain rule,  ddx(ln⁡(ln⁡x))=1ln⁡x⋅d(ln⁡x)dx  ddx(ln⁡(ln⁡x))=1ln⁡x⋅1x  Therefore,  ddx(ln⁡(ln⁡x))=1xln⁡x

Mason Hall · Accepted Answer

The derivative of the natural logarithm is defined to be

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

which has a restriction x>0 that follows from the domain of logarithmic function which is the set of positive real numbers.
To solve for the derivative, use the differentiation rules to simplify.

\frac{d}{d x} (\ln (\ln x)) = \frac{d}{d x} (\ln (\ln x)) \frac{d}{d x} \ln x

by chain rule

= \frac{1}{\ln x} \cdot \frac{d}{d x} \ln x

= \frac{1}{\ln x} \cdot \frac{1}{x}

= \frac{1}{x \ln x}

Therefore,

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

Since the argument of logarithmic functions only includes positive real numbers and in

\ln (\ln x)

the argument is

\ln x

which, as another logarithmic function, has an another argument which is
x
Therefore, the interval on which the result is valid is

(0, \infty)

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Derivatives Evaluate the following derivatives. \frac{d}{dx}(\ln (\ln x))

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