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Beliebt Rechnen >

derivative of (-1/(x^2+y^2))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

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

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

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

integral of \sqrt[3]{x^8}

\int 3 x^{8} d x

(dy)/(dx)=0.3*y

\frac{d y}{d x} = 0.3 \cdot y

integral of 1/(ssqrt((4s)^2-9))

\int \frac{1}{s ( 4 s ) ^{2} - 9} d s

(\partial}{\partial x}(\frac{3r+s)/t)

\frac{\partial}{\partial x} (\frac{3 r + s}{t})

tangent y=sin(x),\at x= pi/4

t an g e n t y = sin (x), at x = \frac{π}{4}