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A linear function is a basic and essential concept in mathematics. It is called 'linear' because its graph is a straight line. Typically, a linear function is expressed in the form of \(f(x) = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. In simpler terms, the slope \(m\) tells us how steep the line is, and the y-intercept \(b\) is the point where the line crosses the y-axis.
In the original exercise, the function provided is \(f(t) = 2 - \frac{2}{3}t\). Here, the linear function has a slope of \(-\frac{2}{3}\) and a y-intercept of 2. This means the line will fall, or decrease, as it moves from left to right due to the negative slope. Linear functions are straightforward when differentiating because their simplest form involves only the variable raised to the first power.

The power rule is one of the most useful tools in calculus for finding derivatives. It states, if you have a function of the form \(f(x) = ax^n\), then its derivative is \(f'(x) = n \cdot ax^{n-1}\). Simply put, the power of the term becomes a coefficient, and the power decreases by one.
Applying the power rule to linear functions like \(-\frac{2}{3}t\), you recognize that the power of \(t\) is 1, so the derivative simplifies to multiplying the coefficient \(-\frac{2}{3}\) by 1. Therefore, differentiating \(-\frac{2}{3}t\) by the power rule gives \(-\frac{2}{3}\). Since there is no variable left, the term becomes constant once differentiated.
This rule makes differentiation quick and easy, particularly for polynomial functions.

The derivative of a constant is one of the simplest rules in differentiation. If a function involves a constant term, like a number on its own (without a variable), then its derivative is zero. This is because constants do not change, and differentiation measures change.
In our exercise, the term 2 is a constant. When differentiating, this term disappears, as its derivative is 0.

• Example: If \(c\) is a constant, then \(f(x) = c\) leads to a derivative \(f'(x) = 0\).
• It applies to any constant term within a function.

Understanding this rule simplifies much of calculus, as constants merely drop away during differentiation, leaving focus on variable terms.