1. **State the problem:** Find the derivative with respect to $x$ of the function $y = \frac{\tan(x)}{x}$. 2. **Recall the formula:** To differentiate a quotient $\frac{u}{v}$, use the quotient rule: $$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$ 3. **Identify $u$ and $v$:** Here, $u = \tan(x)$ and $v = x$. 4. **Compute derivatives:** - $\frac{du}{dx} = \sec^2(x)$ - $\frac{dv}{dx} = 1$ 5. **Apply the quotient rule:** $$\frac{dy}{dx} = \frac{x \cdot \sec^2(x) - \tan(x) \cdot 1}{x^2} = \frac{x \sec^2(x) - \tan(x)}{x^2}$$ 6. **Interpretation:** The derivative is the difference of $x$ times the derivative of $\tan(x)$ and $\tan(x)$ times the derivative of $x$, all over $x^2$. 7. **Final answer:** $$\frac{dy}{dx} = \frac{x \sec^2(x) - \tan(x)}{x^2}$$ This matches option b.