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

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

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

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

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

Solution steps

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

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

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

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

1 \cdot (x + y + z) - 1 \cdot y = x + z

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

\int_{- 1}^{1} e^{x} d x

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\int_{0}^{1} \frac{1 + x}{1 + x ^{2}} d x

y = x^{5} sin^{2} (2 x)

\int_{- 2}^{2} \frac{1}{4 + x ^{2}} d x

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