91Ó°ÊÓ

The derivative is a cornerstone of Differential Calculus. It measures how a function changes as its input changes—essentially describing the rate of change or the function's slope at a given point. For a function like \( f(x) = x - e^{x} + \tan \left(\frac{2\pi}{7}\right) \), the derivative \( f'(x) \) can tell us about how the function behaves across different values of \( x \).
To find \( f'(x) \), we differentiate each part:

• Derivative of \( x \) is 1 since it increases by 1 for each unit increase in \( x \).
• Derivative of \( e^{x} \) is \( e^{x} \). This exponential component grows rapidly, influencing the overall behavior significantly.
• Since \( \tan\left(\frac{2\pi}{7}\right) \) is a constant, its derivative is 0.

By combining these, we get \( f'(x) = 1 - e^{x} \). This derivative helps us understand whether the function is increasing or decreasing, depending on the sign of \( f'(x) \).

An interval of increase for a function refers to the range of \( x \) values where the function is rising. If \( f'(x) > 0 \) over an interval, then \( f(x) \) increases within that interval. In our example, determining these intervals is crucial to understanding the behavior of the function \( f(x) = x - e^{x} + \tan\left(\frac{2\pi}{7}\right) \).
We found that the derivative \( f'(x) = 1 - e^{x} \) should be greater than zero to signify an interval of increase. This results in the inequality \( 1 - e^{x} > 0 \), which simplifies further to \( e^{x} < 1 \).
By taking the natural logarithm of both sides, we have \( x < \ln(1) \), meaning \( x < 0 \), since \( \ln(1) = 0 \). Therefore, the function increases in the interval \((-\infty, 0)\). Recognizing this interval helps identify where the function is rising in value.

Exponential functions feature prominently in calculus and have a unique property: the rate of growth is proportional to the function's current value. The function \( e^{x} \) is a classic example, frequently showing in calculus due to its distinctive property where the derivative is equal to the function itself, \( e^{x} \).
This exponential growth plays a significant role in our example, \( f(x) = x - e^{x} + \tan\left(\frac{2\pi}{7}\right) \), affecting the function's behavior more rapidly as \( x \) increases.
Understanding \( e^{x} \) is essential. Unlike polynomial functions, exponential functions grow without bound at a faster rate, making them more impactful on derivatives and resulting mathematical behavior.
In practical terms, evaluating \( e^{x} \) for positive \( x \) shows rapid increase, while \( e^{x} \) for negative \( x \) approaches zero, hence impacting the signs of expressions like \( 1 - e^{x} \) significantly and altering where \( f(x) \) increases.