derivative csc^7(2t^pi+pi^3)

解答

导数

csc^{7} (2 t^{π} + π^{3})

- 14 π t^{π - 1} csc^{7} (2 t^{π} + π^{3}) cot (2 t^{π} + π^{3})

求解步骤

\frac{d}{d t} (csc^{7} (2 t^{π} + π^{3}))

使用连锁律:

7 (csc (2 t^{π} + π^{3}))^{6} \frac{d}{d t} (csc (2 t^{π} + π^{3}))

= 7 (csc (2 t^{π} + π^{3}))^{6} \frac{d}{d t} (csc (2 t^{π} + π^{3}))

\frac{d}{d t} (csc (2 t^{π} + π^{3})) = - cot (2 t^{π} + π^{3}) csc (2 t^{π} + π^{3}) \cdot 2 π t^{π - 1}

= 7 (csc (2 t^{π} + π^{3}))^{6} (- cot (2 t^{π} + π^{3}) csc (2 t^{π} + π^{3}) \cdot 2 π t^{π - 1})

化简

7 csc^{6} (2 t^{π} + π^{3}) (- cot (2 t^{π} + π^{3}) csc (2 t^{π} + π^{3}) \cdot 2 π t^{π - 1}) : - 14 π t^{π - 1} csc^{7} (2 t^{π} + π^{3}) cot (2 t^{π} + π^{3})

= - 14 π t^{π - 1} csc^{7} (2 t^{π} + π^{3}) cot (2 t^{π} + π^{3})