(\partial)/(\partial y)(3arctan(x/y))

Lösung

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

- \frac{3 x}{x ^{2} + y ^{2}}

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

= 3 \frac{\partial}{\partial y} (arctan (\frac{x}{y}))

Wende die Kettenregel an:

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

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

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

= 3 \cdot \frac{1}{( \frac{x}{y} ) ^{2} + 1} (- \frac{x}{y ^{2}})

Vereinfache

3 \cdot \frac{1}{( \frac{x}{y} ) ^{2} + 1} (- \frac{x}{y ^{2}}) : - \frac{3 x}{x ^{2} + y ^{2}}

= - \frac{3 x}{x ^{2} + y ^{2}}

\frac{\partial}{\partial y} (- sin (x y) y)

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

x \to 0 lim (x^{2} - x^{4})

\frac{\partial}{\partial x} (6 x y - 2 + 10 x^{2} y^{2})

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