What is the derivative of-(ln(2)/(x(x+ln(2)))) ?

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

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

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

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العربية

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

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

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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