What is (\partial)/(\partial z)(yarctan(x/z)) ?

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

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(\partial)/(\partial z)(yarctan(x/z))

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

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

Solution steps

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

Treat

y, x

as constants

Take the constant out:

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

= y \frac{\partial}{\partial z} (arctan (\frac{x}{z}))

Apply the chain rule:

\frac{1}{( \frac{x}{z} ) ^{2} + 1} \frac{\partial}{\partial z} (\frac{x}{z})

= \frac{1}{( \frac{x}{z} ) ^{2} + 1} \frac{\partial}{\partial z} (\frac{x}{z})

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

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

Simplify

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

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

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