What is (\partial)/(\partial x)((4x+y)/(x-y)) ?

The answer to (\partial)/(\partial x)((4x+y)/(x-y)) is -(5y)/((x-y)^2)

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(\partial)/(\partial x)((4x+y)/(x-y))

\frac{\partial}{\partial x} (\frac{4 x + y}{x - y})

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

Solution steps

\frac{\partial}{\partial x} (\frac{4 x + y}{x - y})

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} ( 4 x + y ) ( x - y ) - \frac{\partial}{\partial x} ( x - y ) ( 4 x + y )}{( x - y ) ^{2}}

\frac{\partial}{\partial x} (4 x + y) = 4

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

= \frac{4 ( x - y ) - 1 \cdot ( 4 x + y )}{( x - y ) ^{2}}

Simplify

\frac{4 ( x - y ) - 1 \cdot ( 4 x + y )}{( x - y ) ^{2}} : - \frac{5 y}{( x - y ) ^{2}}

= - \frac{5 y}{( x - y ) ^{2}}

x \to - \infty lim (3^{4 x})

\int_{0}^{0.5} 5 arccos (x) d x

y = \frac{1}{x ^{2}} + x, x = 2, x = 4

x \to 0 lim (x^{6} sin (x))

f (x) = e^{x} - 2 x

What is (\partial)/(\partial x)((4x+y)/(x-y)) ?
The answer to (\partial)/(\partial x)((4x+y)/(x-y)) is -(5y)/((x-y)^2)