(\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}}

d er i v a t i v e f (x) = \frac{2 x}{1 + x ^{2}}

\frac{\partial}{\partial x} (\frac{4 x ^{3}}{x ^{2} + y ^{2}})

\frac{d y}{d x} = 9 y e^{x + 9}

\frac{\partial}{\partial x} (\frac{1}{2 x})

x \to 8 lim (- (2 x - 8))