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Beliebt Rechnen >

derivative f(x)=7arctan(x-sqrt(1+x^2))

Lösung

ableitung von

f (x) = 7 arctan (x - 1 + x^{2})

Lösung

\frac{7 ( 1 + x ^{2} - x )}{( 2 x ^{2} - 2 x x ^{2} + 1 + 2 ) 1 + x ^{2}}

Schritte zur Lösung

\frac{d}{d x} (7 arctan (x - 1 + x^{2}))

Entferne die Konstante:

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

= 7 \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}}

= 7 \cdot \frac{1}{( x - 1 + x ^{2} ) ^{2} + 1} (1 - \frac{x}{1 + x ^{2}})

Vereinfache

7 \cdot \frac{1}{( x - 1 + x ^{2} ) ^{2} + 1} (1 - \frac{x}{1 + x ^{2}}) : \frac{7 ( 1 + x ^{2} - x )}{( 2 x ^{2} - 2 x x ^{2} + 1 + 2 ) 1 + x ^{2}}

= \frac{7 ( 1 + x ^{2} - x )}{( 2 x ^{2} - 2 x x ^{2} + 1 + 2 ) 1 + x ^{2}}

Graph

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