Calculate the derivative of the following functions.
y = csc (t² + t)

Verified step by step guidance

1. Step 1: Identify the outer function and the inner function. Here, the outer function is \( \csc(u) \) and the inner function is \( u = t^2 + t \).
2. Step 2: Differentiate the outer function \( \csc(u) \) with respect to \( u \). The derivative of \( \csc(u) \) is \( -\csc(u)\cot(u) \).
3. Step 3: Differentiate the inner function \( u = t^2 + t \) with respect to \( t \). The derivative is \( 2t + 1 \).
4. Step 4: Apply the chain rule. Multiply the derivative of the outer function by the derivative of the inner function: \( \frac{dy}{dt} = -\csc(t^2 + t)\cot(t^2 + t) \cdot (2t + 1) \).
5. Step 5: Simplify the expression if possible to get the final derivative form.