Derivative of sqrt(x)*sin(2*x)

1. Apply the product rule:

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

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

   1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

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

   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:

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

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

2. Now simplify:

   $$\frac{4 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{2 \sqrt{x}}$$

The answer is:

$$\frac{4 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{2 \sqrt{x}}$$