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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

\int \frac{x ^{2} - 3 x - 7}{( 2 x + 3 ) ( x + 1 )} d x

\int_{1}^{e^{3}} \frac{1}{x ( 1 + ln ( x ) )} d x

y^{^{''}} + 25 y = 4 csc^{2} (5 t)

\frac{\partial}{\partial x} (3 x cos (y))

x \to 0 lim (\frac{1}{cos ( 8 x )})