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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int x (x + 3)^{2} d x

(3 x^{2} + 3 x - 1)^{4}

10 \frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} + 6 y = 0

n = 2 \sum \infty \frac{2}{n ^{2} + 4 n + 3}

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