We can think of the derivative as the slope of  a curve at a single point, but we can also think of it as a rate of change.

If given the position of an object x at time t, we can represent that as the function x(t). The derivative of x(t) will give us the velocity at time time, expressed by the function v(t).  It's easy to see how the derivative of the position is the velocity when thinking about rate of change. If an object is still at a certain position and we apply a rate of change, the object moves at a certain velocity. Now if we change the velocity, either increase or decrease, that is acceleration. Therefore the derivative of v(t) gives use the new function a(t), which is the acceleration at time t.

Suppose an object moves along a coordinate line, its position at each time is given by x(t), given that t is greater than or equal to 0.


If x(t) = 4t^3 + 12t^2 - 6t + 7, what is the position, velocity and acceleration at time to.
to = 6

The position at    to = 6 is x(6) = 4(6)^3 + 12(6)^2 - 6(6) + 7 = 1267

The velocity function v(t) is obtained by taking the derivative of x(t), which is 

12t^2 + 24t - 6, therefore v(6) = 12(6)^2 + 24(6) - 6 = 570

The acceleration function a(t) is obtained by taking the derivative of v(t), which is

24t + 24, therefore a(6) = 24(6) + 24 = 168.