What is (\partial)/(\partial x)((x-y)/(x^2+y^2+1)) ?

The answer to (\partial)/(\partial x)((x-y)/(x^2+y^2+1)) is (-x^2+2xy+y^2+1)/((x^2+y^2+1)^2)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(\partial)/(\partial x)((x-y)/(x^2+y^2+1))

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

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

Solution steps

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

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

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

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

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

Simplify

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

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

f (x) = \frac{6 x + 14}{x + 7}

\int \frac{5}{sec ( x ) - 1} d x

\int \frac{1}{4 x - x ^{2}} d x

x^{2} - 4 \cdot x, x \cdot 0, [- 2, 4]

\frac{4}{x ^{6}} + \frac{2}{x} - \frac{5}{x ^{4}}

What is (\partial)/(\partial x)((x-y)/(x^2+y^2+1)) ?
The answer to (\partial)/(\partial x)((x-y)/(x^2+y^2+1)) is (-x^2+2xy+y^2+1)/((x^2+y^2+1)^2)