# The Derivative

In order to understand the response of the basic R, L, and C elements to a sinusoidal signal, you need to examine the concept of the derivative in some detail. It will not be necessary that you become proficient in the mathematical technique, but simply that you understand the impact of a relationship defined by a derivative.
The derivative dx/dt is defined as the rate of change of x with respect to time. If x fails to change at a particular instant, dx = 0, and the derivative is zero.
For the sinusoidal waveform, $dx/dt$ is zero only at the positive and negative peaks ($wt = \pi/2 \text{and} wt =3\pi/2$ in Fig. 1), since $x$ fails to change at these instants of time. The derivative $dx/dt$ is actually the slope of the graph at any instant of time. Fig. 1: Defining those points in a sinusoidal waveform that have maximum and minimum derivatives. Fig. 2: Derivative of the sine wave of Fig. 1.
A close examination of the sinusoidal waveform will also indicate that the greatest change in $x$ will occur at the instants $wt = 0, \pi, \text{ and } 2\pi$. The derivative is therefore a maximum at these points. At 0 and $2\pi$, x increases at its greatest rate, and the derivative is given a positive sign since $x$ increases with time. At $\pi$, $dx/dt$ decreases at the same rate as it increases at $0$ and $2\pi$, but the derivative is given a negative sign since $x$ decreases with time. Since the rate of change at 0, $\pi$, and $2\pi$ is the same, the magnitude of the derivative at these points is the same also. For various values of $wt$ between these maxima and minima, the derivative will exist and will have values from the minimum to the maximum inclusive. A plot of the derivative in Fig. 2 shows that the derivative of a sine wave is a cosine wave. The peak value of the cosine wave is directly related to the frequency of the original waveform. The higher the frequency, the steeper the slope at the horizontal axis and the greater the value of $dx/dt$, as shown in Fig. 3 for two different frequencies. Fig. 3: Effect of frequency on the peak value of the derivative
Note in Fig. 3 that even though both waveforms (x1 and x2) have the same peak value, the sinusoidal function with the higher frequency produces the larger peak value for the derivative. In addition, note that
the derivative of a sine wave has the same period and frequency as the original sinusoidal waveform.
For the sinusoidal voltage $$e(t) = E_m \sin(wt+\theta)$$ the derivative can be found directly by differentiation (calculus) to produce the following: $$\begin{split} d \over dt {e(t)} &= w E_m \cos(wt+\theta) \\ &=2\pi f E_m \cos(wt+\theta) \end{split}$$ The mechanics of the differentiation process will not be discussed or investigated here; nor will they be required to continue with the text. Note, however, that the peak value of the derivative, $2pfE_m$, is a function of the frequency of $e(t)$, and the derivative of a sine wave is a cosine wave.