What is the derivative of (x*sin(x)/(1+x^2)) ?

The derivative of (x*sin(x)/(1+x^2)) is (x^3cos(x)-x^2sin(x)+xcos(x)+sin(x))/((1+x^2)^2)

What is the first derivative of (x*sin(x)/(1+x^2)) ?

The first derivative of (x*sin(x)/(1+x^2)) is (x^3cos(x)-x^2sin(x)+xcos(x)+sin(x))/((1+x^2)^2)

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

\frac{x ^{3} cos ( x ) - x ^{2} sin ( x ) + x cos ( x ) + sin ( x )}{( 1 + x ^{2} ) ^{2}}

Solution steps

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

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

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

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

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

Simplify

\frac{( sin ( x ) + x cos ( x ) ) ( 1 + x ^{2} ) - 2 xx sin ( x )}{( 1 + x ^{2} ) ^{2}} : \frac{x ^{3} cos ( x ) - x ^{2} sin ( x ) + x cos ( x ) + sin ( x )}{( 1 + x ^{2} ) ^{2}}

= \frac{x ^{3} cos ( x ) - x ^{2} sin ( x ) + x cos ( x ) + sin ( x )}{( 1 + x ^{2} ) ^{2}}

\int_{0}^{π} 1 - cos^{2} (x) d x

\frac{d}{d x} (8 e^{π x} + 2 e^{7 x})

\frac{\partial}{\partial x} (x cos (y) - y e^{x})

n = 1 \sum \infty \frac{5}{n}

\int e x^{2} d x

What is the derivative of (x*sin(x)/(1+x^2)) ?
The derivative of (x*sin(x)/(1+x^2)) is (x^3cos(x)-x^2sin(x)+xcos(x)+sin(x))/((1+x^2)^2)
What is the first derivative of (x*sin(x)/(1+x^2)) ?
The first derivative of (x*sin(x)/(1+x^2)) is (x^3cos(x)-x^2sin(x)+xcos(x)+sin(x))/((1+x^2)^2)