Derivative of y=csc(3x²+1)

1. Rewrite the function to be differentiated:

   $$\csc{\left(3 x^{2} + 1 \right)} = \frac{1}{\sin{\left(3 x^{2} + 1 \right)}}$$

2. Let $u = \sin{\left(3 x^{2} + 1 \right)}$.

3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x^{2} + 1 \right)}$:

   1. Let $u = 3 x^{2} + 1$.

   2. The derivative of sine is cosine:

      $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x^{2} + 1\right)$:

      1. Differentiate $3 x^{2} + 1$ term by term:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x^{2}$ goes to $2 x$

            So, the result is: $6 x$

         2. The derivative of the constant $1$ is zero.

         The result is: $6 x$

      The result of the chain rule is:

      $$6 x \cos{\left(3 x^{2} + 1 \right)}$$

   The result of the chain rule is:

   $$- \frac{6 x \cos{\left(3 x^{2} + 1 \right)}}{\sin^{2}{\left(3 x^{2} + 1 \right)}}$$

5. Now simplify:

   $$- \frac{6 x \cos{\left(3 x^{2} + 1 \right)}}{\sin^{2}{\left(3 x^{2} + 1 \right)}}$$

The answer is:

$$- \frac{6 x \cos{\left(3 x^{2} + 1 \right)}}{\sin^{2}{\left(3 x^{2} + 1 \right)}}$$