#### Answer

$y'=-3x^{1/2}(x+1)^2csc(x+1)^3cot(x+1)^3+\frac{1}{2}x^{-1/2}csc(x+1)^3$

#### Work Step by Step

Rewrite the equation: $y=x^{1/2}csc(x+1)^3$
Take the derivative of the equation using Power Rule, Product Rule, Trigonometric derivatives, and Chain Rule:
$y'=x^{1/2}\times-csc(x+1)^3cot(x+1)^3\times3(x+1)^2\times1+\frac{1}{2}x^{-1/2}csc(x+1)^3$
$=-3x^{1/2}(x+1)^2csc(x+1)^3cot(x+1)^3+\frac{1}{2}x^{-1/2}csc(x+1)^3$