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

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

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

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

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

Solution steps

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

Treat

z, y

as constants

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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