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}})

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

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

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

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

Simplify

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

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

\frac{d}{d x} (e^{5 x} \cdot 5)

u + 1 (1 - 8 u^{2})^{7}

e^{- b t}

x \to - 2 - lim (\frac{1 - x}{x ^{3} + 8})

\frac{d}{d x} ((1 + i)^{x})

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)