What is (\partial)/(\partial y)((y^3)/(x^2+y^2)) ?

The answer to (\partial)/(\partial y)((y^3)/(x^2+y^2)) is (3y^2x^2+y^4)/((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)((y^3)/(x^2+y^2))

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

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

Solution steps

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

\frac{\partial}{\partial y} (y^{3}) = 3 y^{2}

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

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

Simplify

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

= \frac{3 y ^{2} x ^{2} + y ^{4}}{( x ^{2} + y ^{2} ) ^{2}}

n = 0 \sum \infty \frac{3 n ^{9}}{n !}

\frac{d}{d x} (3 cos (x) + 3 sin (x))

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

\int (x + 3)^{\frac{1}{2}} d x

\frac{d y}{d x} 7 x^{3} + 7 y^{3} - 5 x y = 0

What is (\partial)/(\partial y)((y^3)/(x^2+y^2)) ?
The answer to (\partial)/(\partial y)((y^3)/(x^2+y^2)) is (3y^2x^2+y^4)/((x^2+y^2)^2)