Derivative of e^sin^2x

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

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

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

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

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

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

      1. The derivative of sine is cosine:

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

      The result of the chain rule is:

      $$2 \sin{\left(x \right)} \cos{\left(x \right)}$$

   The result of the chain rule is:

   $$2 e^{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}$$

4. Now simplify:

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

The answer is:

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