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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

f (x) = cos (7 x)

\frac{x ^{2}}{5 ( 1 - x )}

\frac{d}{d x} (cos (x t))

x \to 0.5 lim (f (x))

\int_{1}^{5} \frac{2}{x ^{3}} d x

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