Derivative of y=root(1+x)sqrt(x+3)

1. Apply the product rule:

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

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

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

   1. Let $u = x + 3$.

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

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

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

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

         2. The derivative of the constant $3$ is zero.

         The result is: $1$

      The result of the chain rule is:

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

   The result is: $\frac{\sqrt{x + 1}}{2 \sqrt{x + 3}} + \frac{\sqrt{x + 3}}{2 \sqrt{x + 1}}$

2. Now simplify:

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

The answer is: