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

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

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(\partial)/(\partial y)(y/(x^2+y^2+z^2))

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

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

Solution steps

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

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

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

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

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

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

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

\int \frac{1}{2 x ^{2} + x} d x

\frac{d}{d x} ((cos (x))^{s i n (x)})

e^{x}, 14 e^{- x} + 5, x = 0

\int x (4 x^{2} + 8) d x

\int x 1 - x^{4} d x

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