What is (\partial)/(\partial x)((5x+4y)/(x+y)) ?

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

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(\partial)/(\partial x)((5x+4y)/(x+y))

\frac{\partial}{\partial x} (\frac{5 x + 4 y}{x + y})

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

Solution steps

\frac{\partial}{\partial x} (\frac{5 x + 4 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} ( 5 x + 4 y ) ( x + y ) - \frac{\partial}{\partial x} ( x + y ) ( 5 x + 4 y )}{( x + y ) ^{2}}

\frac{\partial}{\partial x} (5 x + 4 y) = 5

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

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

Simplify

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

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

\frac{\partial}{\partial x} (e^{x yz - 8})

(x + 1)^{2}

y^{^{'}} = \frac{2 x + sec ^{2} ( x )}{2 y}, y (0) = - 7

\frac{d}{d x} (2^{(2 - x^{4})})

\frac{d y}{d x} = \frac{x ^{3} + 2 y ^{3}}{x y ^{2}}

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