derivative of (x^3+ix^2+x+1/(x^2+i))

Lösung

\frac{d}{d x} (\frac{x ^{3} + i x ^{2} + x + 1}{x ^{2} + i})

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

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{3} + i x ^{2} + x + 1}{x ^{2} + i})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( x ^{3} + i x ^{2} + x + 1 ) ( x ^{2} + i ) - \frac{d}{d x} ( x ^{2} + i ) ( x ^{3} + i x ^{2} + x + 1 )}{( x ^{2} + i ) ^{2}}

\frac{d}{d x} (x^{3} + i x^{2} + x + 1) = (3 x^{2} + 1) + 2 i x

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

= \frac{( 3 x ^{2} + 1 + 2 i x ) ( x ^{2} + i ) - 2 x ( x ^{3} + i x ^{2} + x + 1 )}{( x ^{2} + i ) ^{2}}

Vereinfache

\frac{( 3 x ^{2} + 1 + 2 i x ) ( x ^{2} + i ) - 2 x ( x ^{3} + i x ^{2} + x + 1 )}{( x ^{2} + i ) ^{2}} : \frac{x ^{4} + 3 i x ^{2} - x ^{2} - 4 x + i}{( x ^{2} + i ) ^{2}}

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

\int \frac{e ^{7 x}}{e ^{7 x} + 8} d x

\int \frac{1}{( y - 1 ) ^{\frac{1}{3}}} d y

\int_{0}^{5} 3 x d x

\int e^{x} \cdot x^{2} d x

a re a - 2 x^{3} + 17 x^{2} - 19 x - 14, [0, x]