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

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

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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

Solution steps

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

Simplify

\frac{- y}{x ^{2} + y ^{2}} : - \frac{y}{x ^{2} + y ^{2}}

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Quotient Rule:

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

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

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

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

Expand

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

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

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