derivative (5x)/(1+x^2)

Lösung

ableitung von

\frac{5 x}{1 + x ^{2}}

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

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\frac{\partial}{\partial x} (x y + x)

\frac{d}{d x} (- 2 sin (x) + 2 sin (2 x))

\int (cos (x) - sec (x)) d x

d er i v a t i v e (t + 5)^{\frac{2}{3}} (3 t^{2} - 2)^{3}

\frac{d y}{d x} + 5 y = 70, y (1) = 110