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

The derivative of (xsin(x)/(1+x)) is (x^2cos(x)+xcos(x)+sin(x))/((1+x)^2)

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

The first derivative of (xsin(x)/(1+x)) is (x^2cos(x)+xcos(x)+sin(x))/((1+x)^2)

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

\frac{x ^{2} cos ( x ) + x cos ( x ) + sin ( x )}{( 1 + x ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (x sin (x)) = sin (x) + x cos (x)

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

= \frac{( sin ( x ) + x cos ( x ) ) ( 1 + x ) - 1 \cdot x sin ( x )}{( 1 + x ) ^{2}}

Simplify

\frac{( sin ( x ) + x cos ( x ) ) ( 1 + x ) - 1 \cdot x sin ( x )}{( 1 + x ) ^{2}} : \frac{x ^{2} cos ( x ) + x cos ( x ) + sin ( x )}{( 1 + x ) ^{2}}

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

\int_{0}^{8} \frac{1}{x ^{2} - 6 x + 5} d x

\frac{d}{d x} (sec (5 x^{3} e^{x}))

x \to 0 lim (6 cos (6 x))

n = 1 \sum \infty \frac{3 ^{n}}{n ! \cdot n}

x \to \infty lim (\frac{1}{x ^{\frac{7}{2}}})

What is the derivative of (xsin(x)/(1+x)) ?
The derivative of (xsin(x)/(1+x)) is (x^2cos(x)+xcos(x)+sin(x))/((1+x)^2)
What is the first derivative of (xsin(x)/(1+x)) ?
The first derivative of (xsin(x)/(1+x)) is (x^2cos(x)+xcos(x)+sin(x))/((1+x)^2)