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

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

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

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

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

Solution steps

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

Treat

z, x

as constants

Take the constant out:

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

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

Apply the Sum/Difference Rule:

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

= z (\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}

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

Simplify

z (0 - 0 - \frac{z}{z ^{2} y ^{2} + 1}) : - \frac{z ^{2}}{z ^{2} y ^{2} + 1}

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

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

\frac{2 ( - x - 1 )}{( x - 1 ) ^{3}}

2 (- \frac{7}{9} + \frac{8}{3} ln (2))

\frac{x}{( x + 7 ) ^{5}}

x^{2 (x^{3} - 5)}

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