derivative of (cos(x)/(x^2+tan(x)))

Lösung

\frac{d}{d x} (\frac{cos ( x )}{x ^{2} + tan ( x )})

\frac{- sin ( x ) ( x ^{2} + tan ( x ) ) - cos ( x ) ( 2 x + sec ^{2} ( x ) )}{( x ^{2} + tan ( x ) ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{cos ( x )}{x ^{2} + tan ( x )})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (cos (x)) = - sin (x)

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

= \frac{( - sin ( x ) ) ( x ^{2} + tan ( x ) ) - ( 2 x + sec ^{2} ( x ) ) cos ( x )}{( x ^{2} + tan ( x ) ) ^{2}}

Vereinfache

= \frac{- sin ( x ) ( x ^{2} + tan ( x ) ) - cos ( x ) ( 2 x + sec ^{2} ( x ) )}{( x ^{2} + tan ( x ) ) ^{2}}