(\partial)/(\partial x)((2x)/(x^2+y^2+z^2))

Lösung

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

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Vereinfache

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

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

x \to + 0 lim (\frac{( 1 + x ) ^{4} - 1}{x})

\frac{\partial}{\partial y} (x y + yz + z x)

x \to 5 lim (\frac{f ( x )}{g ( x )})

\int_{0}^{2} 6 - x^{2} d x

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