Derivative of log5x+19log9x

1. Differentiate $\log{\left(5 x \right)} + 19 \log{\left(9 x \right)}$ term by term:

   1. Let $u = 5 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$\frac{1}{x}$$

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

      1. Let $u = 9 x$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 9 x$:

         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: $9$

         The result of the chain rule is:

         $$\frac{1}{x}$$

      So, the result is: $\frac{19}{x}$

   The result is: $\frac{20}{x}$

The answer is:

$$\frac{20}{x}$$