Derivative of y=sin2x*e²x+1

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

   1. Apply the product rule:

      $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

      $f{\left(x \right)} = e^{2} \sin{\left(2 x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

         1. Let $u = 2 x$.

         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 $\frac{d}{d x} 2 x$:

            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:

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

         So, the result is: $2 e^{2} \cos{\left(2 x \right)}$

      $g{\left(x \right)} = x$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

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

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

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

2. Now simplify:

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

The answer is:

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