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

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

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

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

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

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

Solution steps

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

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

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

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

Expand

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

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

f (x, y) = 9 - (x^{2} + y^{2} - 4)^{2}

f (x, y) = x^{2} y - x^{2} - 2 y^{2}

f (x) = x^{3} - 3 x + 2

f (x) = - \frac{cos ( 2 x )}{2} - 2 sin (x)

f (x, y) = x^{3} + y^{3} - 3 x y

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