Derivative of 2*x-1+1/(x+1)

1. Differentiate $\left(2 x - 1\right) + \frac{1}{x + 1}$ term by term:

   1. Differentiate $2 x - 1$ 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: $2$

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

      The result is: $2$

   2. Let $u = x + 1$.

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

   4. 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:

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

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

2. Now simplify:

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

The answer is:

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