CALCULUSHigher Concepts of Mathematicsf(x) = ae^{bx}(5-9)d [f(x)]dxabe^{bx}These general techniques for finding the derivatives of functions are important for those whoperform detailed mathematical calculations for dynamic systems. For example, the designers ofnuclear facility systems need an understanding of these techniques, because these techniques arenot encountered in the day-to-day operation of a nuclear facility. As a result, the operators ofthese facilities should understand what derivatives are in terms of a rate of change and a slopeof a graph, but they will not normally be required to find the derivatives of functions.The notation d[f(x)]/dxis the common way to indicate the derivative of a function. In someapplications, the notation is used. In other applications, the so-called dot notation is usedf(x)to indicate the derivative of a function with respect to time. For example, the derivative of theamount of heat transferred, Q, with respect to time, dQ/dt, is often written as .QIt is also of interest to note that many detailed calculations for dynamic systems involve not onlyone derivative of a function, but several successive derivatives. The second derivative of afunction is the derivative of its derivative; the third derivative is the derivative of the secondderivative, and so on. For example, velocity is the first derivative of distance traveled withrespect to time, v = ds/dt; acceleration is the derivative of velocity with respect to time, a = dv/dt.Thus, acceleration is the second derivative of distance traveled with respect to time. This iswritten as d^{2}s/dt^{2}. The notation d^{2}[f(x)]/dx^{2} is the common way to indicate the second derivativeof a function. In some applications, the notation is used. The notation for third, fourth,f(x)and higher order derivatives follows this same format. Dot notation can also be used for higherorder derivatives with respect to time. Two dots indicates the second derivative, three dots thethird derivative, and so on.ApplicationofDerivativestoPhysicalSystemsThere are many different problems involving dynamic physical systems that are most readilysolved using derivatives. One of the best illustrations of the application of derivatives areproblems involving related rates of change. When two quantities are related by some knownphysical relationship, their rates of change with respect to a third quantity are also related. Forexample, the area of a circle is related to its radius by the formula . If for some reasonA pr^{2}the size of a circle is changing in time, the rate of change of its area, with respect to time forexample, is also related to the rate of change of its radius with respect to time. Although theseapplications involve finding the derivative of function, they illustrate why derivatives are neededto solve certain problems involving dynamic systems.MA-05 Page 38Rev. 0