d/(da)((cos^2(a))/(sin(a)-1))

Lösung

\frac{d}{d a} (\frac{cos ^{2} ( a )}{sin ( a ) - 1})

\frac{- sin ( 2 a ) ( sin ( a ) - 1 ) - cos ^{3} ( a )}{( sin ( a ) - 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d a} (\frac{cos ^{2} ( a )}{sin ( a ) - 1})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d a} ( cos ^{2} ( a ) ) ( sin ( a ) - 1 ) - \frac{d}{d a} ( sin ( a ) - 1 ) cos ^{2} ( a )}{( sin ( a ) - 1 ) ^{2}}

\frac{d}{d a} (cos^{2} (a)) = - sin (2 a)

\frac{d}{d a} (sin (a) - 1) = cos (a)

= \frac{( - sin ( 2 a ) ) ( sin ( a ) - 1 ) - cos ( a ) cos ^{2} ( a )}{( sin ( a ) - 1 ) ^{2}}

Vereinfache

\frac{( - sin ( 2 a ) ) ( sin ( a ) - 1 ) - cos ( a ) cos ^{2} ( a )}{( sin ( a ) - 1 ) ^{2}} : \frac{- sin ( 2 a ) ( sin ( a ) - 1 ) - cos ^{3} ( a )}{( sin ( a ) - 1 ) ^{2}}

= \frac{- sin ( 2 a ) ( sin ( a ) - 1 ) - cos ^{3} ( a )}{( sin ( a ) - 1 ) ^{2}}