Derivative of sin^4(3x^2-2)

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

2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

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

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

   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 $\frac{d}{d x} \left(3 x^{2} - 2\right)$:

      1. Differentiate $3 x^{2} - 2$ 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 $\left(-1\right) 2$ is zero.

         The result is: $6 x$

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$24 x \sin^{3}{\left(3 x^{2} - 2 \right)} \cos{\left(3 x^{2} - 2 \right)}$$

4. Now simplify:

   $$24 x \sin^{3}{\left(3 x^{2} - 2 \right)} \cos{\left(3 x^{2} - 2 \right)}$$

The answer is:

$$24 x \sin^{3}{\left(3 x^{2} - 2 \right)} \cos{\left(3 x^{2} - 2 \right)}$$