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

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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

\int \frac{x - 1}{x ^{2} - 2 x} d x

\frac{d}{d t} (2)

\int x + x e^{x} d x

x^{3} + x^{2} - 6 x, 6 x

f (x) = x^{2} - 4, [- 4, 3]

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