Derivative of 10x-ln(x+10)^10

1. Differentiate $10 x - \log{\left(x + 10 \right)}^{10}$ 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: $10$

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

      1. Let $u = \log{\left(x + 10 \right)}$.

      2. Apply the power rule: $u^{10}$ goes to $10 u^{9}$

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

         1. Let $u = x + 10$.

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

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

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

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

               2. The derivative of the constant $10$ is zero.

               The result is: $1$

            The result of the chain rule is:

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

         The result of the chain rule is:

         $$\frac{10 \log{\left(x + 10 \right)}^{9}}{x + 10}$$

      So, the result is: $- \frac{10 \log{\left(x + 10 \right)}^{9}}{x + 10}$

   The result is: $10 - \frac{10 \log{\left(x + 10 \right)}^{9}}{x + 10}$

2. Now simplify:

   $$\frac{10 \left(x - \log{\left(x + 10 \right)}^{9} + 10\right)}{x + 10}$$

The answer is:

$$\frac{10 \left(x - \log{\left(x + 10 \right)}^{9} + 10\right)}{x + 10}$$