What is (\partial)/(\partial y)(x-z-arctan(yz)) ?

The answer to (\partial)/(\partial y)(x-z-arctan(yz)) is -z/(z^2y^2+1)

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(\partial)/(\partial y)(x-z-arctan(yz))

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

- \frac{z}{z ^{2} y ^{2} + 1}

Solution steps

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

Treat

x, z

as constants

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

\frac{\partial}{\partial y} (x) = 0

\frac{\partial}{\partial y} (z) = 0

\frac{\partial}{\partial y} (arctan (yz)) = \frac{z}{z ^{2} y ^{2} + 1}

= 0 - 0 - \frac{z}{z ^{2} y ^{2} + 1}

Simplify

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

What is (\partial)/(\partial y)(x-z-arctan(yz)) ?
The answer to (\partial)/(\partial y)(x-z-arctan(yz)) is -z/(z^2y^2+1)