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

The answer to (\partial)/(\partial x)(-(xy)/((1+x)^3)) is -(y(1-2x))/((1+x)^4)

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

= - \frac{y ( 1 - 2 x )}{( 1 + x ) ^{4}}

y = x (1.1)

\frac{d}{d x} (x^{3} - 8)

x \to 0 lim (\frac{( x - 1 ) ^{3} + 1}{x})

\int_{0}^{8} e^{\frac{x}{2}} d x

f (x) = x + \frac{49}{x}

What is (\partial)/(\partial x)(-(xy)/((1+x)^3)) ?
The answer to (\partial)/(\partial x)(-(xy)/((1+x)^3)) is -(y(1-2x))/((1+x)^4)