Acceleration is Second Derivative of Displacement with respect to Time

Theorem
The acceleration $\mathbf a$ of a body $M$ is the second derivative of the displacement $\mathbf s$ of $M$ from a given point of reference time $t$:


 * $\mathbf a = \dfrac {\d^2 \mathbf s} {\d t^2}$

Proof
By definition, the acceleration of a body $M$ is defined as the first derivative of the velocity $\mathbf v$ of $M$ relative to a given point of reference time:


 * $\mathbf a = \dfrac {\d \mathbf v} {\d t}$

Also by definition, the velocity of $M$ is defined as the first derivative of the displacement of $M$ from a given point of reference time:


 * $\mathbf v = \dfrac {\d \mathbf s} {\d t}$

That is:


 * $\mathbf a = \map {\dfrac \d {\d t} } {\dfrac {\d \mathbf s} {\d t} }$

Hence the result by definition of the second derivative.