Derivative of sin(x)*e^2*x

1. Apply the product rule:

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

   $f{\left(x \right)} = e^{2} \sin{\left(x \right)}$; 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. The derivative of sine is cosine:

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

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

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

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

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

2. Now simplify:

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

The answer is:

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