derivative y=(csc(x^2))/(7x-cos^2(x))

解

導関数

y = \frac{csc ( x ^{2} )}{7 x - cos ^{2} ( x )}

\frac{- 2 x cot ( x ^{2} ) csc ( x ^{2} ) ( 7 x - cos ^{2} ( x ) ) - csc ( x ^{2} ) ( 7 + sin ( 2 x ) )}{( 7 x - cos ^{2} ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{csc ( x ^{2} )}{7 x - cos ^{2} ( x )})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( csc ( x ^{2} ) ) ( 7 x - cos ^{2} ( x ) ) - \frac{d}{d x} ( 7 x - cos ^{2} ( x ) ) csc ( x ^{2} )}{( 7 x - cos ^{2} ( x ) ) ^{2}}

\frac{d}{d x} (csc (x^{2})) = - cot (x^{2}) csc (x^{2}) \cdot 2 x

\frac{d}{d x} (7 x - cos^{2} (x)) = 7 + sin (2 x)

= \frac{( - cot ( x ^{2} ) csc ( x ^{2} ) \cdot 2 x ) ( 7 x - cos ^{2} ( x ) ) - ( 7 + sin ( 2 x ) ) csc ( x ^{2} )}{( 7 x - cos ^{2} ( x ) ) ^{2}}

簡素化

= \frac{- 2 x cot ( x ^{2} ) csc ( x ^{2} ) ( 7 x - cos ^{2} ( x ) ) - csc ( x ^{2} ) ( 7 + sin ( 2 x ) )}{( 7 x - cos ^{2} ( x ) ) ^{2}}