1. The problem asks to find the derivative of $\tan(ax)$ with respect to $x$. 2. According to the derivative formulas given, the derivative of $\tan(ax)$ is: $$\frac{d}{dx} (\tan(ax)) = a \sec^2(ax)$$ 3. Here, $a$ is a constant multiplier inside the function, and the chain rule applies. 4. The formula means you multiply the derivative of the inside function $ax$ (which is $a$) by the derivative of $\tan$ which is $\sec^2$ of the inside function. 5. So the final answer is: $$\boxed{a \sec^2(ax)}$$ This completes the solution for problem number 3.