Derivative of x+log(4*x)-1

1. Differentiate $\left(x + \log{\left(4 x \right)}\right) - 1$ term by term:

   1. Differentiate $x + \log{\left(4 x \right)}$ term by term:

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

      2. Let $u = 4 x$.

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

      4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$:

         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: $4$

         The result of the chain rule is:

         $$\frac{1}{x}$$

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

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

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

2. Now simplify:

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

The answer is:

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