What is the derivative of 1/(x(5+ln(x))) ?

The derivative of 1/(x(5+ln(x))) is -(ln(x)+6)/(x^2(5+ln(x))^2)

What is the first derivative of 1/(x(5+ln(x))) ?

The first derivative of 1/(x(5+ln(x))) is -(ln(x)+6)/(x^2(5+ln(x))^2)

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derivative of

\frac{1}{x ( 5 + ln ( x ) )}

- \frac{ln ( x ) + 6}{x ^{2} ( 5 + ln ( x ) ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{1}{x ( 5 + ln ( x ) )})

Apply exponent rule:

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

= \frac{d}{d x} ((x (5 + ln (x)))^{- 1})

Apply the chain rule:

- \frac{1}{( x ( 5 + ln ( x ) ) ) ^{2}} \frac{d}{d x} (x (5 + ln (x)))

= - \frac{1}{( x ( 5 + ln ( x ) ) ) ^{2}} \frac{d}{d x} (x (5 + ln (x)))

\frac{d}{d x} (x (5 + ln (x))) = ln (x) + 6

= - \frac{1}{( x ( 5 + ln ( x ) ) ) ^{2}} (ln (x) + 6)

Simplify

- \frac{1}{( x ( 5 + ln ( x ) ) ) ^{2}} (ln (x) + 6) : - \frac{ln ( x ) + 6}{x ^{2} ( 5 + ln ( x ) ) ^{2}}

= - \frac{ln ( x ) + 6}{x ^{2} ( 5 + ln ( x ) ) ^{2}}

What is the derivative of 1/(x(5+ln(x))) ?
The derivative of 1/(x(5+ln(x))) is -(ln(x)+6)/(x^2(5+ln(x))^2)
What is the first derivative of 1/(x(5+ln(x))) ?
The first derivative of 1/(x(5+ln(x))) is -(ln(x)+6)/(x^2(5+ln(x))^2)