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Beliebt Rechnen >

(\partial)/(\partial x)(1/((x^2+y^2+z^2)))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Wende Exponentenregel an:

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

= \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

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

Vereinfache

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

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

Beliebte Beispiele

(dy)/(dt)= 3/t y+t^5

\frac{d y}{d t} = \frac{3}{t} y + t^{5}

limit as x approaches 8 of-6x

x \to 8 lim (- 6 x)

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

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

integral of (-2x-1)/((x^2+x+1)^6)

\int \frac{- 2 x - 1}{( x ^{2} + x + 1 ) ^{6}} d x

(dy)/(dx)=(y^2)/(x^2)

\frac{d y}{d x} = \frac{y ^{2}}{x ^{2}}