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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Simplify

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

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

7 e^{x}

\frac{d x}{d t} = k (60 - x)

\int x 1 + x x d x

f (x) = (2 x - 1)^{2}

\int \frac{1}{\frac{1}{u} + 2} d u

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