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

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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