What is the derivative of ln(1+ln(x)) ?

The derivative of ln(1+ln(x)) is 1/(x(1+ln(x)))

What is the first derivative of ln(1+ln(x)) ?

The first derivative of ln(1+ln(x)) is 1/(x(1+ln(x)))

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

\frac{1}{x ( 1 + ln ( x ) )}

Solution steps

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

Apply the chain rule:

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

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

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

= \frac{1}{1 + ln ( x )} \cdot \frac{1}{x}

Simplify

\frac{1}{1 + ln ( x )} \cdot \frac{1}{x} : \frac{1}{x ( 1 + ln ( x ) )}

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

What is the derivative of ln(1+ln(x)) ?
The derivative of ln(1+ln(x)) is 1/(x(1+ln(x)))
What is the first derivative of ln(1+ln(x)) ?
The first derivative of ln(1+ln(x)) is 1/(x(1+ln(x)))