When you take the derivative of a function continuously, you get the first derivative, second derivative, third derivative, fourth derivative, fifth derivative, and so on. The first derivative is the result of differentiating a function once. The second derivative means that you have differentiated the first derivative and so it is called the second derivative of the original function. Similarly the third derivative means that you have differentiated the second derivative of a function.
Higher order derivatives are thus obtained when a function is differentiated again and again.
As you know, velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. Now we also know that a derivative of a function is the rate of change of that function. Thus velocity is the derivative of displacement and acceleration is the derivative of velocity. Further, acceleration is the second derivative of displacement.
So suppose that the function f(t) = t^3 + 4t^2 + t + 3 represents the distance traveled by an object in time ‘t’, then its velocity is given by its first derivative:
Velocity or first derivative, f `(t) = 3t^2 + 8t + 1
Then acceleration of the object is given by the second derivative of its distance function (which is obtained by differentiating the velocity function or first derivative)
Acceleration or second derivative, f ``(t) = 6t + 8