This research represents an organized comparison of the efficiency of classical iterative root-finding algorithms like Newton-Raphson, secant and bisection with Ramanujan technique which is more specifically designed for finding differentiating minimum positive roots of algebraic and non-linear equations. Classical techniques are fairly generalized methods and can be applied almost everywhere, yet these have high sensitivity with regards to the choice of initial value or interval and can provide any type of roots rather than being specifically related to the smallest one. On the contrary, the technique used by Ramanujan relies on a unique expansion series and a recurrence relationship and ensures a precise and systematic process towards the closest root which becomes quite useful in practical applications where early detection and identification is involved such as in environmental monitoring and early detection of medical conditions. Through practical examples based on real life application, this research highlights the efficacy of Ramanujan technique in particular instances where the earliest root needs to be obtained. It will become quite clear that for finding the smallest root the Ramanujan approach is more appropriate whereas classical algorithms can be considered in situations where any root suffices or function irregularities are involved.