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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

s im pl i f y x y

d er i v a t i v e \frac{- 2 x - y}{x + 2 y}

\frac{d ^{4}}{d x ^{4}} (sin (x^{2}))

\int (x - \frac{1}{x})^{2} \cdot x d x

\frac{d}{d x} (e^{c o s h (8 x)})

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