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

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

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

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

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

Solution steps

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

Treat

x, y

as constants

Apply the Quotient Rule:

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

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

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

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

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

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

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

\frac{d y}{d x} + 15 x^{2} y = 45 x^{2}

\frac{d}{d x} (\frac{1}{1 - 5 x})

\int \frac{5}{x} + 5 x d x

\frac{d y}{d x} = 4 (x^{2} + 1)

\frac{\partial}{\partial x} (5 x + 6 y)

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