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

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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

y^{^{'}} = arctan (\frac{x}{2})

\int x \cdot e^{- 5 x} d x

\frac{d}{d x} (5^{5 x})

t \to 0 lim (e^{- 7 t} i + \frac{t ^{2}}{sin ^{2} ( t )} j + tan (3 t) k)

\int \frac{1}{3} tan (3 x) d x

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