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Beliebt Rechnen >

(\partial)/(\partial x)(y/(x^2+y^2+z^2))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

steigung y=sqrt(7x+1),(5,6)

s l o p e y = 7 x + 1, (5, 6)

(\partial)/(\partial x)(xye^{x-2z})

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

(\partial)/(\partial x)(A+B*cos(x)sin(y))

\frac{\partial}{\partial x} (A + B \cdot cos (x) sin (y))

(dx)/(dt)=x^{2/3}

\frac{d x}{d t} = x^{\frac{2}{3}}

inverselaplace 1/(4s^2)

in v erse l a pl a ce \frac{1}{4 s ^{2}}