What is the derivative of 1/(x*ln(x)) ?

The derivative of 1/(x*ln(x)) is -(ln(x)+1)/(x^2ln^2(x))

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

The first derivative of 1/(x*ln(x)) is -(ln(x)+1)/(x^2ln^2(x))

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

- \frac{ln ( x ) + 1}{x ^{2} ln ^{2} ( x )}

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{1}{( x ln ( x ) ) ^{2}} \frac{d}{d x} (x ln (x))

= - \frac{1}{( x ln ( x ) ) ^{2}} \frac{d}{d x} (x ln (x))

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

= - \frac{1}{( x ln ( x ) ) ^{2}} (ln (x) + 1)

Simplify

- \frac{1}{( x ln ( x ) ) ^{2}} (ln (x) + 1) : - \frac{ln ( x ) + 1}{x ^{2} ln ^{2} ( x )}

= - \frac{ln ( x ) + 1}{x ^{2} ln ^{2} ( x )}

What is the derivative of 1/(x*ln(x)) ?
The derivative of 1/(x*ln(x)) is -(ln(x)+1)/(x^2ln^2(x))
What is the first derivative of 1/(x*ln(x)) ?
The first derivative of 1/(x*ln(x)) is -(ln(x)+1)/(x^2ln^2(x))