Derivative of 6x-ln(x+5)^6+3

1. Differentiate $\left(6 x - \log{\left(x + 5 \right)}^{6}\right) + 3$ term by term:

   1. Differentiate $6 x - \log{\left(x + 5 \right)}^{6}$ 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: $6$

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

         1. Let $u = \log{\left(x + 5 \right)}$.

         2. Apply the power rule: $u^{6}$ goes to $6 u^{5}$

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + 5 \right)}$:

            1. Let $u = x + 5$.

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

            3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 5\right)$:

               1. Differentiate $x + 5$ term by term:

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

                  2. The derivative of the constant $5$ is zero.

                  The result is: $1$

               The result of the chain rule is:

               $$\frac{1}{x + 5}$$

            The result of the chain rule is:

            $$\frac{6 \log{\left(x + 5 \right)}^{5}}{x + 5}$$

         So, the result is: $- \frac{6 \log{\left(x + 5 \right)}^{5}}{x + 5}$

      The result is: $6 - \frac{6 \log{\left(x + 5 \right)}^{5}}{x + 5}$

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

   The result is: $6 - \frac{6 \log{\left(x + 5 \right)}^{5}}{x + 5}$

2. Now simplify:

   $$\frac{6 \left(x - \log{\left(x + 5 \right)}^{5} + 5\right)}{x + 5}$$

The answer is:

$$\frac{6 \left(x - \log{\left(x + 5 \right)}^{5} + 5\right)}{x + 5}$$