Derivative of 15x-2ln(x-3)^3+6

1. Differentiate $\left(15 x - 2 \log{\left(x - 3 \right)}^{3}\right) + 6$ term by term:

   1. Differentiate $15 x - 2 \log{\left(x - 3 \right)}^{3}$ 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: $x$ goes to $1$

         So, the result is: $15$

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

         1. Let $u = \log{\left(x - 3 \right)}$.

         2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

            1. Let $u = x - 3$.

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

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

               1. Differentiate $x - 3$ term by term:

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

                  2. The derivative of the constant $-3$ is zero.

                  The result is: $1$

               The result of the chain rule is:

               $$\frac{1}{x - 3}$$

            The result of the chain rule is:

            $$\frac{3 \log{\left(x - 3 \right)}^{2}}{x - 3}$$

         So, the result is: $- \frac{6 \log{\left(x - 3 \right)}^{2}}{x - 3}$

      The result is: $15 - \frac{6 \log{\left(x - 3 \right)}^{2}}{x - 3}$

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

   The result is: $15 - \frac{6 \log{\left(x - 3 \right)}^{2}}{x - 3}$

2. Now simplify:

   $$\frac{3 \left(5 x - 2 \log{\left(x - 3 \right)}^{2} - 15\right)}{x - 3}$$

The answer is:

$$\frac{3 \left(5 x - 2 \log{\left(x - 3 \right)}^{2} - 15\right)}{x - 3}$$