d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}})

Lösung

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

- \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

= - \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

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

= - \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 x

Vereinfache

- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 x : - \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

= - \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

\frac{\partial}{\partial x} (x e^{x^{2} y})

d er i v a t i v e \frac{- 40}{x ^{7}}

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

\frac{d}{d x} (- \frac{1}{x + 1})

\frac{d}{d s} (s L (y (s)))