To find the derivative of the function $y = \log_{10}(2x)$, we can apply the chain rule. We start by letting $u = 2x$, which allows us to rewrite the function as $y = \log_{10}(u)$.

By applying the chain rule, we get:

Next, we need to determine $\frac{dy}{du}$. The derivative of a logarithmic function is given by the formula:

In this case, since $a = 10$, we have:

Now, substituting back for $u$, we find:

Next, we compute $\frac{du}{dx}$:

Now, substituting $\frac{dy}{du}$ and $\frac{du}{dx}$ back into our chain rule equation, we have:

Simplifying this expression, we find:

Thus, the derivative of $y = \log_{10}(2x)$ is: