Derivative of e^(sin(1-3*x)^(2))

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

2. The derivative of $e^{u}$ is itself.

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

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

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

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

      1. Let $u = 1 - 3 x$.

      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 $\frac{d}{d x} \left(1 - 3 x\right)$:

         1. Differentiate $1 - 3 x$ term by term:

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

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

               1. Apply the power rule: $x$ goes to $1$

               So, the result is: $-3$

            The result is: $-3$

         The result of the chain rule is:

         $$- 3 \cos{\left(3 x - 1 \right)}$$

      The result of the chain rule is:

      $$- 6 \sin{\left(1 - 3 x \right)} \cos{\left(3 x - 1 \right)}$$

   The result of the chain rule is:

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

4. Now simplify:

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

The answer is:

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