Derivative of y=sinx^2+sin2x+sin2^2

1. Differentiate $\left(\sin^{2}{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin^{2}{\left(2 \right)}$ term by term:

   1. Differentiate $\sin^{2}{\left(x \right)} + \sin{\left(2 x \right)}$ term by term:

      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:

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

      4. Let $u = 2 x$.

      5. The derivative of sine is cosine:

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

      6. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

         1. 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: $2$

         The result of the chain rule is:

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

      The result is: $2 \sin{\left(x \right)} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}$

   2. The derivative of the constant $\sin^{2}{\left(2 \right)}$ is zero.

   The result is: $2 \sin{\left(x \right)} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}$

2. Now simplify:

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

The answer is:

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