Derivative of sqrt(x+1)-(1/x)

1. Differentiate $\sqrt{x + 1} - \frac{1}{x}$ term by term:

   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:

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

   4. The derivative of a constant times a function is the constant times the derivative of the function.

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

      So, the result is: $\frac{1}{x^{2}}$

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

2. Now simplify:

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

The answer is:

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