What is the derivative of (ln(x)/(1+ln(2x))) ?

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

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

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

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