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

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

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

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

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

Solution steps

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

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

\frac{\partial}{\partial y} (x + 2 y) = 2

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

Simplify

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

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

\frac{\partial}{\partial x} (y^{4} sin (4 x))

y^{^{''}} y^{^{'}} = 2

\int_{0}^{\infty} 10 e^{- 5 x} d x

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

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

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