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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d x}{d x} = 1

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

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

Simplify

\frac{1 \cdot ( x ^{2} + y ^{2} ) - ( 2 x + 2 y \frac{d y}{d x} ) x}{( x ^{2} + y ^{2} ) ^{2}} : \frac{- x ^{2} - 2 x y \frac{d y}{d x} + y ^{2}}{( x ^{2} + y ^{2} ) ^{2}}

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

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