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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\int 4 x^{2} + x^{4} d x

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\int \frac{3 x ^{2} + 1}{x ^{2} \cdot 4 - x ^{2}} d x

\frac{\partial}{\partial x} (sin (x \cdot y \cdot e^{x^{2}}))

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