1. Problem: Find $\frac{dy}{dx}$ if $y = \ln 5x$. 2. Formula: The derivative of $y = \ln u$ with respect to $x$ is $\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}$. 3. Here, $u = 5x$. So, $\frac{du}{dx} = 5$. 4. Applying the formula: $$\frac{dy}{dx} = \frac{1}{5x} \cdot 5 = \frac{5}{5x} = \frac{1}{x}$$ 5. Explanation: The natural logarithm function's derivative is the reciprocal of its argument times the derivative of the argument. Since $5x$ is a linear function, its derivative is 5, which cancels with the 5 in the denominator. 6. Final answer: $$\boxed{\frac{dy}{dx} = \frac{1}{x}}$$