derivative of (2y/(3x^2+2y^2))

Lösung

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

\frac{2 ( 3 x ^{2} \frac{d y}{d x} - 6 x y - 2 y ^{2} \frac{d y}{d x} )}{( 3 x ^{2} + 2 y ^{2} ) ^{2}}

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (3 x^{2} + 2 y^{2}) = 6 x + 4 y \frac{d y}{d x}

= 2 \cdot \frac{\frac{d y}{d x} ( 3 x ^{2} + 2 y ^{2} ) - ( 6 x + 4 y \frac{d y}{d x} ) y}{( 3 x ^{2} + 2 y ^{2} ) ^{2}}

Vereinfache

2 \cdot \frac{\frac{d y}{d x} ( 3 x ^{2} + 2 y ^{2} ) - ( 6 x + 4 y \frac{d y}{d x} ) y}{( 3 x ^{2} + 2 y ^{2} ) ^{2}} : \frac{2 ( 3 x ^{2} \frac{d y}{d x} - 6 x y - 2 y ^{2} \frac{d y}{d x} )}{( 3 x ^{2} + 2 y ^{2} ) ^{2}}

= \frac{2 ( 3 x ^{2} \frac{d y}{d x} - 6 x y - 2 y ^{2} \frac{d y}{d x} )}{( 3 x ^{2} + 2 y ^{2} ) ^{2}}

y^{^{''}} + 14 y^{^{'}} + 85 y = 0

d er i v a t i v e \frac{3 x + 2}{4 x}

x \to 3 lim (x + 3 x - 3)

\int_{- 3}^{0} x 1 - x d x

\int 6 x e^{x} d x