derivative ((cos^3(x)-1))/(cos^2(x))

解

導関数

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

\frac{- 3 cos ^{4} ( x ) sin ( x ) + sin ( 2 x ) ( cos ^{3} ( x ) - 1 )}{cos ^{4} ( x )}

解答ステップ

\frac{d}{d x} (\frac{cos ^{3} ( x ) - 1}{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} ( cos ^{3} ( x ) - 1 ) cos ^{2} ( x ) - \frac{d}{d x} ( cos ^{2} ( x ) ) ( cos ^{3} ( x ) - 1 )}{( cos ^{2} ( x ) ) ^{2}}

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

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

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

簡素化

\frac{( - 3 cos ^{2} ( x ) sin ( x ) ) cos ^{2} ( x ) - ( - sin ( 2 x ) ) ( cos ^{3} ( x ) - 1 )}{( cos ^{2} ( x ) ) ^{2}} : \frac{- 3 cos ^{4} ( x ) sin ( x ) + sin ( 2 x ) ( cos ^{3} ( x ) - 1 )}{cos ^{4} ( x )}

= \frac{- 3 cos ^{4} ( x ) sin ( x ) + sin ( 2 x ) ( cos ^{3} ( x ) - 1 )}{cos ^{4} ( x )}