derivative of (2xy^2/((1+2x^2)^2))

Lösung

\frac{d}{d x} (\frac{2 x y ^{2}}{( 1 + 2 x ^{2} ) ^{2}})

\frac{2 y ( 4 x ^{3} \frac{d y}{d x} - 6 x ^{2} y + 2 x \frac{d y}{d x} + y )}{( 1 + 2 x ^{2} ) ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{2 x y ^{2}}{( 1 + 2 x ^{2} ) ^{2}})

Entferne die Konstante:

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

= 2 \frac{d}{d x} (\frac{x y ^{2}}{( 1 + 2 x ^{2} ) ^{2}})

Wende die Quotientenregel an:

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

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

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

\frac{d}{d x} ((1 + 2 x^{2})^{2}) = 8 x (1 + 2 x^{2})

= 2 \cdot \frac{( y ^{2} + 2 x y \frac{d y}{d x} ) ( 1 + 2 x ^{2} ) ^{2} - 8 x ( 1 + 2 x ^{2} ) x y ^{2}}{( ( 1 + 2 x ^{2} ) ^{2} ) ^{2}}

Vereinfache

2 \cdot \frac{( y ^{2} + 2 x y \frac{d y}{d x} ) ( 1 + 2 x ^{2} ) ^{2} - 8 x ( 1 + 2 x ^{2} ) x y ^{2}}{( ( 1 + 2 x ^{2} ) ^{2} ) ^{2}} : \frac{2 y ( 4 x ^{3} \frac{d y}{d x} - 6 x ^{2} y + 2 x \frac{d y}{d x} + y )}{( 1 + 2 x ^{2} ) ^{3}}

= \frac{2 y ( 4 x ^{3} \frac{d y}{d x} - 6 x ^{2} y + 2 x \frac{d y}{d x} + y )}{( 1 + 2 x ^{2} ) ^{3}}

e x p an d \frac{s ( s ^{2} + 4 )}{( s - 1 ) ( s ^{2} + 8 s + 15 )}

s im pl i f y (x + 7 y^{4}) (6 x - 7 y^{4})

e x p an d \frac{[ K x ( x + 1 ) ]}{x ( x + 1 ) ^{2} + 1}

e x p an d - \frac{8 sin ( θ )}{( 2 - cos ( θ ) ) ^{2}}

e x p an d \frac{4 s + 1}{s ( 2 s + 1 ) ( s + 1 )}