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The derivative of a function is a key concept in calculus. It represents the rate of change of a function as its input varies. To find the derivative, we use its formal definition. The derivative of a function \( f(x) \) at a point \( x \) is defined as:\[ f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \]This formula essentially calculates the change in the function, \( f(x) \), over a very small interval around \( x \). The expression inside the limit, \( \frac{f(x+h) - f(x)}{h} \), is called the difference quotient. As \( h \)—the interval length—approaches zero, the difference quotient approaches the derivative at \( x \).

This limit process provides a precise mathematical way of describing how a function behaves around any point on its curve. The derivative tells us whether the function is increasing or decreasing at a specific point and gives insights into the function's overall behavior.

The limit process is an essential part of understanding derivatives. When we talk about limits in calculus, we're exploring what happens to a function as its input gets closer and closer to a particular value. In the context of derivatives, we are interested in what happens as the change in our input variable, \( h \), tends to zero.

The limit process in our derivative formula, \( \lim_{h \to 0} \), means we're considering values of \( h \) that are getting infinitesimally small, effectively zero. This allows us to see the behavior of the function exactly at point \( x \), without actually dividing by zero. In practice, it provides a way to calculate the exact slope of the tangent line to the curve of \( f(x) \) at any given point.

• For each small value of \( h \), evaluate the difference quotient \( \frac{f(x+h) - f(x)}{h} \).
• As \( h \to 0 \), observe how this quotient behaves.

This is how we harness the power of limits to define derivatives, granting us powerful tools for understanding change and motion.

Simplifying expressions is a pivotal step when calculating derivatives. It transforms complex algebraic expressions into simpler ones, making them easier to evaluate, especially within a limit.

For the function \( f(x) = \frac{x}{2} \), when we substitute the function into the definition of the derivative:\[ f(x+h) = \frac{x+h}{2} \],
we set up a difference quotient:\[ \frac{\frac{x+h}{2} - \frac{x}{2}}{h} \].
To simplify it:

• First, combine like terms in the numerator: \( \frac{x+h}{2} - \frac{x}{2} = \frac{h}{2} \). Here, the \( x \) terms cancel out.
• Using this simplified expression, our limit becomes \( \lim_{h \to 0} \frac{\frac{h}{2}}{h} \).
• Simplify further to \( \lim_{h \to 0} \frac{1}{2} = \frac{1}{2} \), since \( \frac{h}{h} = 1 \).

By breaking down the steps and simplifying each part of the expression, we can quickly and accurately evaluate limits and find derivatives. This process highlights the beauty of calculus in transforming complicated expressions into elegant solutions using basic algebraic principles.