What is (\partial)/(\partial x)((-y-xy)/((1+x)^3)) ?

The answer to (\partial)/(\partial x)((-y-xy)/((1+x)^3)) is (2y)/((x+1)^3)

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(\partial)/(\partial x)((-y-xy)/((1+x)^3))

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

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

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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What is (\partial)/(\partial x)((-y-xy)/((1+x)^3)) ?
The answer to (\partial)/(\partial x)((-y-xy)/((1+x)^3)) is (2y)/((x+1)^3)