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Beliebt Rechnen >

(\partial)/(\partial x)((4x^3)/(x^2+y^2))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

= 4 \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}}

= 4 \cdot \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

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

Vereinfache

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

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

Beliebte Beispiele

(dy)/(dx)=sqrt(9y)e^{x+9}

\frac{d y}{d x} = 9 y e^{x + 9}

(\partial)/(\partial x)(1/(2sqrt(x)))

\frac{\partial}{\partial x} (\frac{1}{2 x})

limit as x approaches 8 of-(2x-8)

x \to 8 lim (- (2 x - 8))

integral of (7x^2-8x)e^{2x}

\int (7 x^{2} - 8 x) e^{2 x} d x

y^{^{'}} = \frac{- y}{x}