What is (\partial)/(\partial x)((-x+3y)/((x+y)^3)) ?

The answer to (\partial)/(\partial x)((-x+3y)/((x+y)^3)) is -(2(5y-x))/((x+y)^4)

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(\partial)/(\partial x)((-x+3y)/((x+y)^3))

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

- \frac{2 ( 5 y - x )}{( x + y ) ^{4}}

Solution steps

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

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

\frac{\partial}{\partial x} (- x + 3 y) = - 1

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

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

Simplify

\frac{( - 1 ) ( x + y ) ^{3} - 3 ( x + y ) ^{2} ( - x + 3 y )}{( ( x + y ) ^{3} ) ^{2}} : - \frac{2 ( 5 y - x )}{( x + y ) ^{4}}

= - \frac{2 ( 5 y - x )}{( x + y ) ^{4}}

\frac{d}{d x} (ln (x) \cdot x)

x (x - 2)^{3}

f (x) = \frac{x}{3 + x}

\int_{0}^{π} cos (x) cos (n x) d x

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

What is (\partial)/(\partial x)((-x+3y)/((x+y)^3)) ?
The answer to (\partial)/(\partial x)((-x+3y)/((x+y)^3)) is -(2(5y-x))/((x+y)^4)