What is (\partial)/(\partial x)((x+y)/(x+z)) ?

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

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(\partial)/(\partial x)((x+y)/(x+z))

\frac{\partial}{\partial x} (\frac{x + y}{x + z})

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

Solution steps

\frac{\partial}{\partial x} (\frac{x + y}{x + z})

Treat

y, z

as constants

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{d x}{d t} = - \frac{x}{10} + \frac{1}{10 e ^{\frac{t}{10}}}

t y^{^{'}} - y = 3 t^{2} cos (2 t)

\int x (8 x + 5)^{8} d x

\int x - \frac{1}{2} d x

y^{^{''}} + 3 y^{^{'}} + 2 y = \frac{1}{2 + e ^{x}}

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