What is the derivative of (xy^2/(x^2+y^2)) ?

The derivative of (xy^2/(x^2+y^2)) is (2x^3y\frac{dy)/(dx)-x^2y^2+y^4}{(x^2+y^2)^2}

What is the first derivative of (xy^2/(x^2+y^2)) ?

The first derivative of (xy^2/(x^2+y^2)) is (2x^3y\frac{dy)/(dx)-x^2y^2+y^4}{(x^2+y^2)^2}

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\frac{d}{d x} (\frac{x y ^{2}}{x ^{2} + y ^{2}})

\frac{2 x ^{3} y \frac{d y}{d x} - x ^{2} y ^{2} + y ^{4}}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{x y ^{2}}{x ^{2} + y ^{2}})

Apply the Quotient Rule:

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

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

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

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

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

Expand

(y^{2} + 2 x y \frac{d y}{d x}) (x^{2} + y^{2}) - (2 x + 2 y \frac{d y}{d x}) x y^{2} : 2 x^{3} y \frac{d y}{d x} - x^{2} y^{2} + y^{4}

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

\int \frac{e ^{x}}{( e ^{x} + 1 )} d x

\int 4 cos (\frac{x}{2}) d x

7 x^{2} + \frac{d y}{d x} (x^{3} + 6) = 0

\int \frac{1}{x - x + 6} d x

\frac{d}{d x} (4 (1 - sin (x)))

What is the derivative of (xy^2/(x^2+y^2)) ?
The derivative of (xy^2/(x^2+y^2)) is (2x^3y\frac{dy)/(dx)-x^2y^2+y^4}{(x^2+y^2)^2}
What is the first derivative of (xy^2/(x^2+y^2)) ?
The first derivative of (xy^2/(x^2+y^2)) is (2x^3y\frac{dy)/(dx)-x^2y^2+y^4}{(x^2+y^2)^2}