(\partial)/(\partial x)(((xv^2))/((x^2+v^2)))

Lösung

\frac{\partial}{\partial x} (\frac{( x v ^{2} )}{( x ^{2} + v ^{2} )})

\frac{v ^{2} ( - x ^{2} + v ^{2} )}{( x ^{2} + v ^{2} ) ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{( x v ^{2} )}{( x ^{2} + v ^{2} )})

Behandle

v

als Konstante

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

= v^{2} \frac{\frac{\partial}{\partial x} ( x ) ( x ^{2} + v ^{2} ) - \frac{\partial}{\partial x} ( x ^{2} + v ^{2} ) x}{( x ^{2} + v ^{2} ) ^{2}}

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

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

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

Vereinfache

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

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

\int \frac{5}{( 5 x + 4 ) ln ( 5 x + 4 )} d x

\int 6 x^{2} - 2 x + 1 d x

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d er i v a t i v e y = 3 2 x

x \to 0 lim (x cot (2 x))