Encyclopedia of Electrical Engineering

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.
**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
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.

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.