derivative y=(x^3+1)/(x^2-1)

Lösung

ableitung von

y = \frac{x ^{3} + 1}{x ^{2} - 1}

\frac{x ( x - 2 )}{( x - 1 ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

d er i v a t i v e f (x) = \frac{x ^{2}}{x ^{2} + 2}

\int 54

d er i v a t i v e (x)^{5 x}

\frac{( x ^{2} + 1 ) d y}{d x} + 3 x (y - 1) = 0

t an g e n t y = \frac{6 x}{x ^{2} + 1}, (1, 3)