A derivative is the rate of change of one quantity with respect to another; for example, the rate at which an object’s velocity changes with respect to time (compare to slope). Such rates of change show up frequently in everyday life. For example, the compound interest law
states that the velocity of interest accumulation is proportional to
the principal account value, given by dV(t)/dt=rV(t) and V(0)=P, where P
is the initial (principal) account value, V(t), a function of time, is
the current account value (on which interest is continuously assessed),
and r is the interest rate (dt is an instantaneous time interval, dV(t)
is the infinitesimal amount by which V(t) changes in this time, and
their quotient is the accumulation rate). Although credit card interest
is typically compounded daily and described by the APR, annual percentage rate,
this differential equation can be solved to give the continuous
solution V(t) = Pe^(rt).
1) Define derivative. Derivative (also called differential quotient; especially British) - the limit of the ratio of the increment of a function (generally y) to the increment of a variable (generally x)
in that function, as the latter tends to 0; the instantaneous change of
one quantity with respect to another, as velocity, which is the instantaneous change of distance with respect to time. Compare first derivative, and second derivative:[1]
- First derivative – the derivative of a function, example: "Velocity is the first derivative of distance with respect to time."
- Second derivative – the derivative of the derivative of a function, example: "Acceleration is the second derivative of distance with respect to time."
2) Know the order and degree of the differential equation. The order of a differential equation is determined by the highest order derivative; the degree is determined by the highest power on a variable. For example, the differential equation shown in Figure 1 is of second-order, third-degree.
3) Know the difference between a general, or complete solution versus a particular solution. A complete solution contains a number of arbitrary constants equal to the order the equation. (To solve an nth order differential equation, you have to perform n
integrations, and each time you integrate, you have to introduce an
arbitrary constant.) For example, in the compound interest law, the
differential equation dy/dt=ky is of order 1, and its complete solution y
= ce^(kt) has exactly 1 arbitrary constant. A particular solution is
obtained by assigning particular values to the constants in the general
solution.
