derivative of ((x^2+1^2)/((x^4+1)^4))

Lösung

\frac{d}{d x} (\frac{( x ^{2} + 1 ) ^{2}}{( x ^{4} + 1 ) ^{4}})

\frac{4 x ( x ^{2} + 1 ) ( - 3 x ^{4} - 4 x ^{2} + 1 )}{( x ^{4} + 1 ) ^{5}}

Schritte zur Lösung

\frac{d}{d x} (\frac{( x ^{2} + 1 ) ^{2}}{( x ^{4} + 1 ) ^{4}})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} ((x^{2} + 1)^{2}) = 4 x (x^{2} + 1)

\frac{d}{d x} ((x^{4} + 1)^{4}) = 16 x^{3} (x^{4} + 1)^{3}

= \frac{4 x ( x ^{2} + 1 ) ( x ^{4} + 1 ) ^{4} - 16 x ^{3} ( x ^{4} + 1 ) ^{3} ( x ^{2} + 1 ) ^{2}}{( ( x ^{4} + 1 ) ^{4} ) ^{2}}

Vereinfache

\frac{4 x ( x ^{2} + 1 ) ( x ^{4} + 1 ) ^{4} - 16 x ^{3} ( x ^{4} + 1 ) ^{3} ( x ^{2} + 1 ) ^{2}}{( ( x ^{4} + 1 ) ^{4} ) ^{2}} : \frac{4 x ( x ^{2} + 1 ) ( - 3 x ^{4} - 4 x ^{2} + 1 )}{( x ^{4} + 1 ) ^{5}}

= \frac{4 x ( x ^{2} + 1 ) ( - 3 x ^{4} - 4 x ^{2} + 1 )}{( x ^{4} + 1 ) ^{5}}

x \to 2 lim (\frac{2 x - 8}{x - 1})

d er i v a t i v e 4 t sin (π t)

x^{3} y^{^{'''}} + 3 x^{2} y^{^{''}} - 2 x y^{^{'}} + 2 y = 0

x \to 5 lim (\frac{6}{x - 5})

\int \frac{sin ( x )}{( cos ( x ) ) ^{3}} d x