Derivative of x*sqrt^4(x+1)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = \left(\sqrt{x + 1}\right)^{4}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \sqrt{x + 1}$.

   2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

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

      1. Let $u = x + 1$.

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

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

         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:

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

      The result of the chain rule is:

      $$2 x + 2$$

   The result is: $x \left(2 x + 2\right) + \left(\sqrt{x + 1}\right)^{4}$

2. Now simplify:

   $$\left(x + 1\right) \left(3 x + 1\right)$$

The answer is:

$$\left(x + 1\right) \left(3 x + 1\right)$$