Explanation:
To differentiate √cosec(x³+1) with respect to x, we need to use the chain rule of differentiation. The chain rule states that if we have a function f(u) and u=g(x), then the derivative of f(u) with respect to x is given by:

dy/dx = dy/du * du/dx

where dy/du is the derivative of f(u) with respect to u and du/dx is the derivative of g(x) with respect to x.

Step 1: Rewrite the function
Before we start, let's rewrite the function as:

√cosec(x³+1) = (cosec(x³+1))^1/2

Step 2: Find the derivatives
Now, let's find the derivatives of the function:

dy/du = 1/2 * (cosec(x³+1))^(-1/2) * (-cosec(x³+1)cot(x³+1) * 3x²)

du/dx = 3x²csc(x³+1)cot(x³+1)

Step 3: Apply the chain rule
Now, let's apply the chain rule:

dy/dx = dy/du * du/dx
= 1/2 * (cosec(x³+1))^(-1/2) * (-cosec(x³+1)cot(x³+1) * 3x²) * 3x²csc(x³+1)cot(x³+1)
= -3x^4cot(x³+1)/(2sqrt(cosec(x³+1))^3)

Conclusion:
Thus, the differentiation of √cosec(x³+1) with respect to x is -3x^4cot(x³+1)/(2sqrt(cosec(x³+1))^3).