What is the derivative of (x^3ln(x)/(x+2)) ?

The derivative of (x^3ln(x)/(x+2)) is (2x^3ln(x)+x^3+6x^2ln(x)+2x^2)/((x+2)^2)

What is the first derivative of (x^3ln(x)/(x+2)) ?

The first derivative of (x^3ln(x)/(x+2)) is (2x^3ln(x)+x^3+6x^2ln(x)+2x^2)/((x+2)^2)

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

\frac{2 x ^{3} ln ( x ) + x ^{3} + 6 x ^{2} ln ( x ) + 2 x ^{2}}{( x + 2 ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (x^{3} ln (x)) = 3 x^{2} ln (x) + x^{2}

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

= \frac{( 3 x ^{2} ln ( x ) + x ^{2} ) ( x + 2 ) - 1 \cdot x ^{3} ln ( x )}{( x + 2 ) ^{2}}

Simplify

\frac{( 3 x ^{2} ln ( x ) + x ^{2} ) ( x + 2 ) - 1 \cdot x ^{3} ln ( x )}{( x + 2 ) ^{2}} : \frac{2 x ^{3} ln ( x ) + x ^{3} + 6 x ^{2} ln ( x ) + 2 x ^{2}}{( x + 2 ) ^{2}}

= \frac{2 x ^{3} ln ( x ) + x ^{3} + 6 x ^{2} ln ( x ) + 2 x ^{2}}{( x + 2 ) ^{2}}

What is the derivative of (x^3ln(x)/(x+2)) ?
The derivative of (x^3ln(x)/(x+2)) is (2x^3ln(x)+x^3+6x^2ln(x)+2x^2)/((x+2)^2)
What is the first derivative of (x^3ln(x)/(x+2)) ?
The first derivative of (x^3ln(x)/(x+2)) is (2x^3ln(x)+x^3+6x^2ln(x)+2x^2)/((x+2)^2)