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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial y} ((x + y)^{3} - 12 x y)

\int x \cdot sin (2 x^{2}) d x

x \to 0 + lim (x^{2} e^{\frac{1}{x}})

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

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

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