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}}

\int_{1}^{3} (x^{3} + \frac{1}{4 x ^{3}}) d x

y = 2 x - 1 + 4 x^{2}, (1, 5)

\int k e^{- x} d x

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

g (x) = \frac{1}{4 x ^{2} + 2 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)