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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Take the constant out:

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

= - 2 \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}}

= - 2 \cdot \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)

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

Simplify

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

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

\int \frac{1}{( x ^{2} + y ^{2} ) ^{3}} d y

\int cos (t^{3}) d t

\frac{d}{d x} (ln (9) x^{6})

\int (- 3 v + 4) d v

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