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

Lösung

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

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

Schritte zur Lösung

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

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

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

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

Vereinfache

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

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

\int 54 - 18 z - 6 z^{2} + 2 z^{3} d z

\int_{0}^{1} 2 x 5^{x} d x

d er i v a t i v e v^{\frac{3}{2}} (v + c e^{x})

\int \frac{6 x}{1 + x ^{4}} d x

\frac{d x}{d t} = 349.344897 (1 - e^{- 0.0916 t})