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Beliebt Rechnen >

(\partial)/(\partial x)(1/(2x)arctan(y/x))

Lösung

\frac{\partial}{\partial x} (\frac{1}{2 x} arctan (\frac{y}{x}))

Lösung

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

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{1}{2 x} arctan (\frac{y}{x}))

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = \frac{1}{x}, g = arctan (\frac{y}{x})

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

integral of 1/(\sqrt[3]{x)-x}

\int \frac{1}{3 x - x} d x

tangent (8x^2+4x+4)/(sqrt(4))

t an g e n t \frac{8 x ^{2} + 4 x + 4}{4}

integral of (e^x+5)

\int (e^{x} + 5) d x

limit as x approaches 3 of 9-x^2

x \to 3 lim (9 - x^{2})

limit as x approaches 0 of (ln(x))^2

x \to 0 lim ((ln (x))^{2})