d/(d{z)}(({x})/(({x)^2+{y}^2+{z}^2)^{3/2}})

Lösung

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

- \frac{3 x z}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d z} (x^{2} + y^{2} + z^{2})

= - \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d z} (x^{2} + y^{2} + z^{2})

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

= x (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 z)

Vereinfache

x (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 z) : - \frac{3 x z}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

= - \frac{3 x z}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

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

\int x^{4} 4 + x^{5} d x

x \to 0 + lim (x^{2} + 3)

\frac{\partial}{\partial x} (9 x sin (8 x^{2} y))

l a pl a ce t r an s f or m e^{3 t} cos (2 t)