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

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

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

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

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

Solution steps

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

Simplify

\frac{- z}{y ^{2} + z ^{2}} : - \frac{z}{y ^{2} + z ^{2}}

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Quotient Rule:

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

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

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

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

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

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

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

\int_{0}^{1} x^{4} e^{2 x} d x

\int \frac{1}{( x ^{2} + 6 ) ^{\frac{3}{2}}} d x

n = 0 \sum \infty 2 x^{n}

\frac{x}{x ^{2} + 36}

x \to 0 lim ((x)^{x})

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