derivative f(x)= x/(sqrt(x^2+1))

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d x}{d x} = 1

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

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

Vereinfache

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

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

\frac{\partial}{\partial x} (8 x (x^{2} + y^{2})^{- \frac{1}{2}})

t an g e n t f (x) = x^{2} + 13, at x = 6

\frac{d y}{d x} = 2 y + 34 x + 52

x \to 0 + lim (lo g_{\frac{1}{e}} (x))

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