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Before starting to learn something, I always wonder whats its application. So would you please give some practical examples of application of Successive differentiation and concepts related to it such as Taylor's series?

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Any time you need a higher derivative, you have to get there through successive differentiation. For example, if $s(t)$ is the position of a particle at time $t$, then $\frac{d^3s(t)}{dt^3}$ is the "jerk". To get it, you have to compute $\frac{ds(t)}{dt}$, $\frac{d^2s(t)}{dt^2}$, and then $\frac{d^3s(t)}{dt^3}$.

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