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

Lösung

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

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

Schritte zur Lösung

\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}}}

\int \frac{1}{( x ^{2} - 100 ) ^{2}} d x

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

a re a x = 0, y = 0, y = \frac{- x ^{2} + 8}{9}

θ \to 0 lim (\frac{θ}{sin ( θ )})

\frac{\partial}{\partial x} ((x - y) (1 - x y))