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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derivative of

\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}}

\int 4^{x} d x

d er i v a t i v e f (x) = 5 x^{2} + 4 x - 10

\int 3 cos (x) + 3 x d x

\frac{\partial}{\partial x} (x e^{x y + 1})

\frac{\partial}{\partial y} (2 x sin (x^{2} y))

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)