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}})

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}}

\frac{d}{d x} (\frac{x}{2 + c x})

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

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

\int_{1}^{\infty} \frac{9}{x ^{2} + 1} d x

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