How acceleration is equal to d^2x / dt^2

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Solution

the definition of a derivative is the change in one variable with respect to another. For instance, velocity is the change in position with respect to time. You could do this experiment and calculate velocity as:

V = (x2 - x1)/(t2 - t1)

now imagine that we start shrinking the distance between our measurements until the distance between them is infinitesimally small. We represent this with 'd' notation. So Velocity becomes:

V = dx/dt

where dx = change in position
and dt = change in time

Acceleration is the change in velocity with respect to time or:

a = dV/dt

but V = dx/dt

dx/dt is called a first derivative, so taking the derivative of this quantity with respect to t again gives us the second derivative, d^2x/dt^2, or in words... the change in the change in position with respect to time.

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