Derivative of (sqrt(2x))(x+1)^3

1. Apply the product rule:

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

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

   1. Let $u = 2 x$.

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

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

      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:

      $$\frac{\sqrt{2}}{2 \sqrt{x}}$$

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

   1. Let $u = x + 1$.

   2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

      $$3 \left(x + 1\right)^{2}$$

   The result is: $3 \sqrt{2} \sqrt{x} \left(x + 1\right)^{2} + \frac{\sqrt{2} \left(x + 1\right)^{3}}{2 \sqrt{x}}$

2. Now simplify:

   $$\frac{\sqrt{2} \left(x + 1\right)^{2} \left(7 x + 1\right)}{2 \sqrt{x}}$$

The answer is:

$$\frac{\sqrt{2} \left(x + 1\right)^{2} \left(7 x + 1\right)}{2 \sqrt{x}}$$