What is the derivative of csc^7(2t^pi+pi^3) ?

The derivative of csc^7(2t^pi+pi^3) is -14pit^{pi-1}csc^7(2t^pi+pi^3)cot(2t^pi+pi^3)

What is the first derivative of csc^7(2t^pi+pi^3) ?

The first derivative of csc^7(2t^pi+pi^3) is -14pit^{pi-1}csc^7(2t^pi+pi^3)cot(2t^pi+pi^3)

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derivative

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

Solution steps

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

Apply the chain rule:

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})

Simplify

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})

What is the derivative of csc^7(2t^pi+pi^3) ?
The derivative of csc^7(2t^pi+pi^3) is -14pit^{pi-1}csc^7(2t^pi+pi^3)cot(2t^pi+pi^3)
What is the first derivative of csc^7(2t^pi+pi^3) ?
The first derivative of csc^7(2t^pi+pi^3) is -14pit^{pi-1}csc^7(2t^pi+pi^3)cot(2t^pi+pi^3)