(\partial)/(\partial x)(1/(xy^2+yx^2))

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

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