Derivative of log10(x+7)^9-9x

1. Differentiate $- 9 x + \left(\frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}\right)^{9}$ term by term:

   1. Let $u = \frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}$.

   2. Apply the power rule: $u^{9}$ goes to $9 u^{8}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(x + 7 \right)}}{\log{\left(10 \right)}}$:

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

         1. Let $u = x + 7$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 7\right)$:

            1. Differentiate $x + 7$ term by term:

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

               2. The derivative of the constant $7$ is zero.

               The result is: $1$

            The result of the chain rule is:

            $$\frac{1}{x + 7}$$

         So, the result is: $\frac{1}{\left(x + 7\right) \log{\left(10 \right)}}$

      The result of the chain rule is:

      $$\frac{9 \log{\left(x + 7 \right)}^{8}}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$$

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

      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: $9$

      So, the result is: $-9$

   The result is: $-9 + \frac{9 \log{\left(x + 7 \right)}^{8}}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$

2. Now simplify:

   $$\frac{9 \left(- \left(x + 7\right) \log{\left(10 \right)}^{9} + \log{\left(x + 7 \right)}^{8}\right)}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$$

The answer is:

$$\frac{9 \left(- \left(x + 7\right) \log{\left(10 \right)}^{9} + \log{\left(x + 7 \right)}^{8}\right)}{\left(x + 7\right) \log{\left(10 \right)}^{9}}$$