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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\int_{0}^{+ oo} x \cdot (x + 1)^{- 2} d x

f (x) = 5 - x

\int_{0}^{3} 1 + x d x

\frac{d}{d x} (\frac{3}{5 x})

\frac{d}{d t} (2 sin (t) + 2)

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