Derivative of sqrt(3*x+1)*sin(2*x)

1. Apply the product rule:

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

   $f{\left(x \right)} = \sqrt{3 x + 1}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 3 x + 1$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 1\right)$:

      1. Differentiate $3 x + 1$ term by term:

         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: $3$

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

         The result is: $3$

      The result of the chain rule is:

      $$\frac{3}{2 \sqrt{3 x + 1}}$$

   $g{\left(x \right)} = \sin{\left(2 x \right)}$; 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: $2 \sqrt{3 x + 1} \cos{\left(2 x \right)} + \frac{3 \sin{\left(2 x \right)}}{2 \sqrt{3 x + 1}}$

2. Now simplify:

   $$\frac{\left(12 x + 4\right) \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)}}{2 \sqrt{3 x + 1}}$$

The answer is:

$$\frac{\left(12 x + 4\right) \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)}}{2 \sqrt{3 x + 1}}$$