Derivative of ln(-x²+6x-4)

1. Let $u = \left(- x^{2} + 6 x\right) - 4$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(- x^{2} + 6 x\right) - 4\right)$:

   1. Differentiate $\left(- x^{2} + 6 x\right) - 4$ term by term:

      1. Differentiate $- x^{2} + 6 x$ 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: $x^{2}$ goes to $2 x$

            So, the result is: $- 2 x$

         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: $6$

         The result is: $6 - 2 x$

      2. The derivative of the constant $-4$ is zero.

      The result is: $6 - 2 x$

   The result of the chain rule is:

   $$\frac{6 - 2 x}{\left(- x^{2} + 6 x\right) - 4}$$

4. Now simplify:

   $$\frac{2 \left(x - 3\right)}{x^{2} - 6 x + 4}$$

The answer is:

$$\frac{2 \left(x - 3\right)}{x^{2} - 6 x + 4}$$