(\partial)/(\partial x)(arccos(x/(sqrt(x^2+y^2))))

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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