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Beliebt Rechnen >

(\partial)/(\partial x)(4arctan(y/x))

Lösung

\frac{\partial}{\partial x} (4 arctan (\frac{y}{x}))

Lösung

- \frac{4 y}{y ^{2} + x ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial x} (4 arctan (\frac{y}{x}))

Behandle

y

als Konstante

Entferne die Konstante:

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

= 4 \frac{\partial}{\partial x} (arctan (\frac{y}{x}))

Wende die Kettenregel an:

\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}}

= 4 \cdot \frac{1}{( \frac{y}{x} ) ^{2} + 1} (- \frac{y}{x ^{2}})

Vereinfache

4 \cdot \frac{1}{( \frac{y}{x} ) ^{2} + 1} (- \frac{y}{x ^{2}}) : - \frac{4 y}{y ^{2} + x ^{2}}

= - \frac{4 y}{y ^{2} + x ^{2}}

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