Notation:

The notation d[f(x)]/dx  is the general way to denotes the derivative  of a function.   In some applications, the notation f'(x) is used.  In other applications, the so-called dot notation is used to denote the derivative of a function with respect to time.  For instance, the derivative of the amount of heat transferred, Q, along with respect to time, dQ/dt, is frequent written as Q.

It is also of interest to remember that several detailed calculations for dynamic systems include not only one derivative of a function, but various successive derivatives. A second derivative of a function is the derivative of its derivative; the third derivative is the derivative of the second derivative, and many more. For instance, velocity is the first derivative of distance traveled with respect to time, v = ds/dt; the derivation acceleration is the derivative of velocity with respect to time, a = dv /dt. Therefore, acceleration is the second derivative of distance traveled with respect to time.  This is written as d²s/dt².  The notation d² [f (x)]/dx² is the common way to indicate the second derivative of a function. In a few applications, the notation f'(x) is used.  The notation for third, fourth, and higher order derivatives follows this similar format.  Dot notation can also be used for higher order derivatives with respect to time.  Two dots denote the second derivative and three dots the third derivative, and so on.