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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

\frac{d}{d x} (x^{3} + e^{2 x})

x \to 2 lim (x^{2} + 3 x - 1)

\frac{\partial}{\partial x} (\frac{4 x - 6 y}{y ^{2} + 2 x ^{2}})

\frac{d}{d x} (sin (ln (cos (x))))

\frac{d}{d x} (\frac{5 x ^{2} - 9 x + 13}{2 x + 4})