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

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

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

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

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derivative of

y = \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}}

\frac{d}{d x} ((x^{3} + 5) (5 x^{2} - 5))

x y^{2} \frac{d y}{d x} = y^{3} - x^{3}, y (1) = 4

cos (3 x^{2})

\frac{d y}{d x} = - 9 y, y (0) = - 5

\frac{8 + 5 x}{x ^{2} + 9}

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