(\partial)/(\partial x)((x^3*y)/(x^2+y^2))

Lösung

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

\frac{y ( x ^{4} + 3 x ^{2} y ^{2} )}{( x ^{2} + y ^{2} ) ^{2}}

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

y \frac{3 x ^{2} ( x ^{2} + y ^{2} ) - 2 x x ^{3}}{( x ^{2} + y ^{2} ) ^{2}} : \frac{y ( x ^{4} + 3 x ^{2} y ^{2} )}{( x ^{2} + y ^{2} ) ^{2}}

= \frac{y ( x ^{4} + 3 x ^{2} y ^{2} )}{( x ^{2} + y ^{2} ) ^{2}}

d er i v a t i v e 2^{n}

\int \frac{cos ^{4} ( x )}{sin ( x )} d x

y^{^{'}} = x e^{x^{2} - 4}

d er i v a t i v e y = ln (x^{3} + 4)

\frac{\partial}{\partial x} (6 x - 2 y)