Derivative of log(x+3)-x

1. Differentiate $- x + \log{\left(x + 3 \right)}$ term by term:

   1. Let $u = x + 3$.

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

   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:

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

   4. 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: $-1$

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

2. Now simplify:

   $$- \frac{x + 2}{x + 3}$$

The answer is:

$$- \frac{x + 2}{x + 3}$$