derivative (x-arctan(x))/(x^2)

Lösung

ableitung von

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

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

Schritte zur Lösung

\frac{d}{d x} (\frac{x - arctan ( x )}{x ^{2}})

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\int_{1}^{3} 2 x^{2} + 2 x + 4 d x

d er i v a t i v e 11 x + 2

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

\int e^{3 x} (3 + 9 x) d x

d er i v a t i v e 3 x + 4