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

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

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

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

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

Solution steps

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

Treat

x, z

as constants

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

x \to - 5 lim (\frac{\frac{5}{x} + 1}{x + 5})

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

e^{30 t}

f (x) = 3 x^{2} + 2 x, at x = - 2

\frac{d}{d t} (sin (6 t))

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