d/(da)((tan^2(a)-sin^2(a))/(tan^2(a)))

解

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

\frac{- sin ( 2 a ) + 2 tan ( a )}{tan ^{2} ( a )}

解答ステップ

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

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

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

= \frac{( 2 tan ( a ) sec ^{2} ( a ) - sin ( 2 a ) ) tan ^{2} ( a ) - 2 tan ( a ) sec ^{2} ( a ) ( tan ^{2} ( a ) - sin ^{2} ( a ) )}{( tan ^{2} ( a ) ) ^{2}}

簡素化

\frac{( 2 tan ( a ) sec ^{2} ( a ) - sin ( 2 a ) ) tan ^{2} ( a ) - 2 tan ( a ) sec ^{2} ( a ) ( tan ^{2} ( a ) - sin ^{2} ( a ) )}{( tan ^{2} ( a ) ) ^{2}} : \frac{- sin ( 2 a ) + 2 tan ( a )}{tan ^{2} ( a )}

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