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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

\frac{1 \cdot ( x ^{2} + 2 x y + y ^{2} ) - ( 2 x + 2 y ) x}{( x ^{2} + 2 x y + y ^{2} ) ^{2}} : \frac{y - x}{( x + y ) ^{3}}

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

\frac{d}{d x} ((4 x - 8)^{2})

\int_{0}^{x} 3 e^{- 3 x} d x

x \to \infty lim (13 x)

\int x^{9} x^{5} + 7 d x

\int \frac{x ^{2} - 8}{x + 3} d x

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