Derivative of arcctg^2*5x*ln(x-4)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = x \operatorname{acot}^{2}{\left(5 \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   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: $\operatorname{acot}^{2}{\left(5 \right)}$

   $g{\left(x \right)} = \log{\left(x - 4 \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = x - 4$.

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

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

      1. Differentiate $x - 4$ term by term:

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

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

         The result is: $1$

      The result of the chain rule is:

      $$\frac{1}{x - 4}$$

   The result is: $\frac{x \operatorname{acot}^{2}{\left(5 \right)}}{x - 4} + \log{\left(x - 4 \right)} \operatorname{acot}^{2}{\left(5 \right)}$

2. Now simplify:

   $$\frac{\left(x + \left(x - 4\right) \log{\left(x - 4 \right)}\right) \operatorname{acot}^{2}{\left(5 \right)}}{x - 4}$$

The answer is: