Derivative of x-ln(x+6)

1. Differentiate $x - \log{\left(x + 6 \right)}$ term by term:

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

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

      1. Let $u = x + 6$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 6\right)$:

         1. Differentiate $x + 6$ term by term:

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

            2. The derivative of the constant $6$ is zero.

            The result is: $1$

         The result of the chain rule is:

         $$\frac{1}{x + 6}$$

      So, the result is: $- \frac{1}{x + 6}$

   The result is: $1 - \frac{1}{x + 6}$

2. Now simplify:

   $$\frac{x + 5}{x + 6}$$

The answer is:

$$\frac{x + 5}{x + 6}$$