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

Lösung

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

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

Wende die Produktregel an:

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

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

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

\frac{\partial}{\partial x} (x) = 1

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

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

Vereinfache

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

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