How do you find the derivative of $g(x)=2csc^8(4x)$ ?

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Answer 1 · 2017-12-20T09:28:33

$dy/dx=-64cot(4x)csc^8(4x)$

Chain rule:
$y=2u^8$
$u=csc(v)$
$v=4x$

$dy/dx=dy/(du)*(du)/(dv)*(dv)/(dx)$

$dy/dx=(16u^7)*(-csc(v)cot(v))*(4)$

Now we replace every $v$ with $4x$ and every $u$ with $csc(v)\rightarrow csc(4x)$ :

$dy/dx=(16csc(4x)^7)*(-csc(4x)cot(4x))*(4)$

and rearrange:

$dy/dx=-64csc(4x)cot(4x)csc^7(4x)$

and one more thing to clean up:

$dy/dx=-64cot(4x)csc^8(4x)$

How do you find the derivative of g(x)=2csc^8(4x)g(x)=2csc^8(4x)?