Derivative of ln(1-10x)

1. Let $u = 1 - 10 x$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - 10 x\right)$:

   1. Differentiate $1 - 10 x$ term by term:

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

      2. 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: $-10$

      The result is: $-10$

   The result of the chain rule is:

   $$- \frac{10}{1 - 10 x}$$

4. Now simplify:

   $$\frac{10}{10 x - 1}$$

The answer is:

$$\frac{10}{10 x - 1}$$