derivative f(x)=(sqrt(x))/(x^3+1)

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\int \frac{3}{x ^{2} + 16} d x

\int 5 x^{- 4} d x

t an g e n t f (x) = \frac{3}{x}, at x = 6

\frac{d}{d x} (5 ln (cos (x)))

x \to 0 lim (2 (\frac{a ^{x} - 1}{x}))