What is (\partial)/(\partial y)((x+y)/(x^2+y^2)) ?

The answer to (\partial)/(\partial y)((x+y)/(x^2+y^2)) is (-y^2-2xy+x^2)/((x^2+y^2)^2)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(\partial)/(\partial y)((x+y)/(x^2+y^2))

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

\frac{\partial}{\partial y} (x + y) = 1

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

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

Simplify

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

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

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

\int (4 - 3 x)^{7} d x

cos (2 (t - \frac{π}{2}))

\int - 6 sin (x) d x

\frac{d y}{d x} - \frac{4}{x} y = y^{5} \frac{1}{x ^{8}}

What is (\partial)/(\partial y)((x+y)/(x^2+y^2)) ?
The answer to (\partial)/(\partial y)((x+y)/(x^2+y^2)) is (-y^2-2xy+x^2)/((x^2+y^2)^2)