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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

x

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int_{0}^{1} x 8 - x^{2} d x

(1 + ln (x) + \frac{y}{x}) d x = (7 - ln (x)) d y

x \to - 7 lim (\frac{1}{x + 7})

[t]

\frac{d}{d x} (ln (e^{6 x}))

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