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

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

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

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

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

Solution steps

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

Simplify

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

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

n = 1 \sum \infty \frac{2}{n ^{8}}

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