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When a rocket blasts off, its thrust increases as fuel burns away, meaning its acceleration changes every millisecond. The constant acceleration equations you might know—like v=u+atv = u + at—completely fail here.
To track an object when forces keep changing, we turn to calculus to find instantaneous velocity and acceleration from a displacement equation.
How do we bridge the gap between where an object is and how violently it is speeding up?

The Motion Ladder

Velocity (vv) is the instantaneous rate of change of displacement (ss) with respect to time (tt): v=dsdtv = \frac{ds}{dt}. Acceleration (aa) is the rate of change of velocity: a=dvdt=d2sdt2a = \frac{dv}{dt} = \frac{d^2s}{dt^2}.
In the late 1660s, Isaac Newton invented calculus precisely to model this relationship, developing fluxions to track how planetary positions shift continuously across time.
To differentiate power terms, we apply the power rule: ddt(c⋅tn)=n⋅c⋅tn−1\frac{d}{dt}(c \cdot t^n) = n \cdot c \cdot t^{n-1}, multiplying by the power and reducing the exponent by 1.
Let's see what happens when we apply these derivatives to a real displacement function.

Finding Velocity and Acceleration Functions

Suppose a particle moves along a straight line with displacement s=2t3−9t2+12ts = 2t^3 - 9t^2 + 12t meters for time t≥0t \ge 0 seconds.
Differentiating displacement gives the velocity function: v=dsdt=6t2−18t+12v = \frac{ds}{dt} = 6t^2 - 18t + 12
Differentiating velocity gives the acceleration function: a=dvdt=12t−18a = \frac{dv}{dt} = 12t - 18