d/(dy)(x/(sqrt(x^2+y^2)))

Lösung

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

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

x \to 2 lim (4 x - 3)

\frac{d}{d x} (sin (\frac{1}{x}))

\int \frac{x e ^{x}}{2} d x

\frac{\partial}{\partial x} (ln (x + 3 y + 7 z))

\int tan^{3} (x) (sec (x))^{\frac{3}{2}} d x