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Understanding the product rule is key when dealing with the derivatives of functions that are products of two or more other functions. In calculus, the product rule is a crucial tool for differentiating expressions where two varying quantities are multiplied together. If you have a function defined as the product of two functions, say, \( u(x) \) and \( v(x) \), the derivative of this function \( f(x) = u(x) \cdot v(x) \) is not merely the product of the derivatives of \( u \) and \( v \). The correct approach, following the product rule, is as follows:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
In our trigonometric function \( f(x) = \ln(x) \cos(x) \), we apply the product rule. This rule guides us to differentiate each function — \( \ln(x) \) and \( \cos(x) \) — separately, and then combine them as demonstrated in the step by step solution:
\[ f'(x) = \ln(x) \times \frac{d}{dx} (\cos(x)) + \cos(x) \times \frac{d}{dx}(\ln(x)) \]
Remember, the product rule is essential when you are faced with a product of functions since it allows you to differentiate correctly instead of just separately differentiating and then multiplying.

Finding critical points is a pivotal process in calculus, particularly when we're interested in identifying the peaks and valleys on a curve — the relative extrema. Critical points occur where the first derivative of a function is either zero or undefined. These points are where the behavior of the function may change from increasing to decreasing, or vice versa.
To find these points, like in our trigonometric function example, you first differentiate the function to get \( f'(x) \). Then, the critical points can be found by setting \( f'(x) = 0 \) or determining where \( f'(x) \) does not exist. In the context of our example, because the equation \( \frac{\cos (x)}{x} - \ln (x) \sin (x) = 0 \) involves trigonometric functions combined with a logarithm, solving it directly may not be possible. Instead, one could use approximation methods or graphing utilities to estimate where the function's derivative equals zero.
Once you've identified the critical points, you next evaluate these points to determine if they correspond to relative maxima, minima, or points of inflection. Having determined the locations of critical points, their nature can be discerned through further analysis or graphing.

In modern calculus, graphing utilities such as calculators and software have become indispensable tools for visualizing functions and their properties. These utilities allow us to plot functions and their derivatives, providing a clear graphical representation of the behavior of functions over intervals.
When determining the relative extrema of functions, particularly when they are complex like trigonometric functions tangled with logarithmic parts, graphing utilities can be an asset. By plotting the function and its derivative, graphing utilities enable us to observe the points at which the derivative crosses the x-axis — these are our critical points. Moreover, by closely examining the graph, we can identify the regions where the function is increasing or decreasing, and where it achieves its highest and lowest points within a given interval.
In our trigonometric function exercise, the graphing utility helps to pinpoint the approximate locations of a local maximum and minimum, since solving the derivative equation analytically is challenging. Students should learn to leverage such tools to complement analytical methods, and to gain intuitive insights into the behavior of complex functions.