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

The answer to (\partial)/(\partial x)((2y)/((x+y)^2)) is -(4y)/((x+y)^3)

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

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

- \frac{4 y}{( x + y ) ^{3}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

2 y (- \frac{2}{( x + y ) ^{3}} \cdot 1) : - \frac{4 y}{( x + y ) ^{3}}

= - \frac{4 y}{( x + y ) ^{3}}

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

y = 3 x, x + y = 4, y = 1

\int_{- 1}^{1} \frac{5 r}{( 4 + r ^{2} ) ^{2}} d r

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

\int \frac{3 x - 2}{5 x ^{2} - 3 x + 2} d x

What is (\partial)/(\partial x)((2y)/((x+y)^2)) ?
The answer to (\partial)/(\partial x)((2y)/((x+y)^2)) is -(4y)/((x+y)^3)