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The limit definition of the derivative is crucial for understanding how we calculate the derivative of a function. It is expressed mathematically as:\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]Essentially, this formula tells us how the function's value changes as we make very small adjustments to the input value, \(x\). By evaluating this limit as \(h\) approaches zero, we determine the slope of the tangent line to the function at any point \(x\). This gives us the derivative, reflecting the rate of change of the function at that exact point.

• "\(f(x+h)\)" represents the function's value when \(x\) is increased by a tiny increment \(h\).
• "\(f(x)\)" is the original value of the function at \(x\).
• The fraction \(\frac{f(x+h) - f(x)}{h}\) represents the average rate of change over this little interval.

This process of finding derivatives using the limit approach is foundational in calculus and helps us understand the behavior of various functions.

A function is a relationship between two sets of numbers or variables. In this case, we are dealing with the function \(f(x) = 3x - 2\). Here, \(f(x)\) expresses a linear relationship between \(x\) and \(f(x)\), where:

• "3x" implies that each unit increase in \(x\) results in an increase in \(f(x)\) by a factor of 3.
• "-2" is a constant adjustment to the output value of \(f(x)\).

This function is linear, meaning its graph is a straight line. The derivative or slope of such a function is constant, which aligns with our expectation that the derivative of \(3x - 2\) is always 3. In essence, the function's behavior is very predictable, as it changes in a straightforward, consistent manner.

Simplification plays a key role when working with derivatives, especially when using the limit definition. Once we've substituted the function into our derivative formula, we arrive at this expression:\[ f'(x) = \lim_{{h \to 0}} \frac{[3(x+h) - 2] - (3x - 2)}{h} \]At this point, simplification involves breaking down the expression inside the brackets:

• Distribute the "3" to both \(x\) and \(h\), which gives us \(3x + 3h - 2\).
• Subtract \(3x - 2\), which leads to the simpler \(3h\).

This process of simplification ensures that we have a manageable expression that can then be reduced further by canceling common factors in subsequent steps. Simplifying expressions is critical for making the limit, and thus the derivative, easy to evaluate.

The algebraic method involves manipulating expressions using algebra to calculate derivatives. After simplifying the expression from the previous step, we are left with:\[ f'(x) = \lim_{{h \to 0}} \frac{3h}{h} \]To compute this limit, we can simplify the expression further. Notice that "\(h\)" appears both in the numerator and the denominator, allowing us to cancel them out:

• This leaves us with the constant value \(3\).

Finally, the expression becomes \(\lim_{{h \to 0}} 3 = 3\), which shows that the derivative of \(3x - 2\) is indeed 3, irrespective of the value of \(x\). The algebraic method provides a systematic approach to understanding how derivatives are calculated, reinforcing how changes in \(x\) influence the rate of change in \(f(x)\).