derivative of 3arctan(x+sqrt(1+x^2))

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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