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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

x, z

als Konstanten

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

sum from n=1 to infinity of 5/(10^n)

n = 1 \sum \infty \frac{5}{1 0 ^{n}}

derivative y=14sqrt(x)

d er i v a t i v e y = 14 x

integral from 1 to 5 of (x^2+2)/(6x-x^2)

\int_{1}^{5} \frac{x ^{2} + 2}{6 x - x ^{2}} d x

fläche y^2-2x=2,x-y=3

a re a y^{2} - 2 x = 2, x - y = 3

integral of 2/(x-1)

\int \frac{2}{x - 1} d x