derivative of x/(x^2+y^2)-y^2 with respect to x

Let's perform the differentiation:

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

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

We differentiate

u1

x

and

v2

x^{2} + y^{2}

using:

\frac{d}{d x} (\frac{u1}{v2}) = \frac{\frac{d}{d x} (u1) \cdot v2 - u1 \cdot \frac{d}{d x} (v2)}{{v2}^{2}}

Let's perform the differentiation

u1

\frac{d}{d x} (x)

1

Let's perform the differentiation

v2

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

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

ℹ

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

2 x

‖

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

0

2 x + 0

2 x

You can evaluate the resulted derivative (

\frac{u1}{v2}

)':

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

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

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

Then you can resume the derivation:

ℹ

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

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

‖

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

0

\frac{- x^{2} + y^{2}}{{(x^{2} + y^{2})}^{2}} + 0

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

▶Derivative:

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