Problem 39 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \underline{\phantom{xxx}} f(x)=\frac{2}{x} $$

Step-by-Step Solution

Answer

The derivative \(f^{\prime}(x)\) is \(\frac{-2}{x^2}\).

Step 1: Understand the 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 means we need to evaluate this limit for \(f(x) = \frac{2}{x}\).

Step 2: Substitute the Function

Substitute \( f(x) = \frac{2}{x} \) into the definition of the derivative. Calculate \( f(x+h) \) as \( \frac{2}{x+h} \). So, you have:\[ f^{\prime}(x) = \lim_{h \to 0} \frac{\frac{2}{x+h} - \frac{2}{x}}{h} \].

Step 3: Simplify the Expression

Combine the fractions in the numerator by finding a common denominator, so:\[ f^{\prime}(x) = \lim_{h \to 0} \frac{\frac{2x - 2(x+h)}{x(x+h)}}{h} = \lim_{h \to 0} \frac{-2h}{hx(x+h)} \].

Step 4: Factor and Simplify the Limit

Cancel out \(h\) from the numerator and denominator (note it is non-zero as \(h \rightarrow 0\)):\[ f^{\prime}(x) = \lim_{h \to 0} \frac{-2}{x(x+h)} \].

Step 5: Evaluate the Limit

Take the limit as \(h \to 0\):\[ f^{\prime}(x) = \frac{-2}{x^2} \].

Key Concepts

Definition of Derivative

The concept of the derivative is foundational in calculus. It measures how a function changes as its input changes. The derivative of a function at a point gives the slope of the tangent line to the function at that point, indicating how steeply the function is rising or falling. This is akin to finding the "instantaneous rate of change" of the function.

Mathematically, the derivative of a function \( f(x) \) at a specific point is defined using the limit process:

• \( f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) is a tiny change in \( x \), essentially approaching zero. The definition ensures we're observing the behavior of the function as close to the point as possible. This is why the derivative is also seen as the function's best linear approximation at that point.

Limit Process

The limit process is essential in the calculation of derivatives. It provides a formal way to handle functions as they approach a specific point. Without limits, we wouldn't be able to compute derivatives rigorously. In the context of derivatives, we examine what happens to the expression \( \frac{f(x+h) - f(x)}{h} \) as \( h \) approaches zero.

This process might seem abstract, but imagine you want to find how fast a car is going at an exact moment. You could calculate it over an incredibly tiny interval to approximate that rate. Similarly, using the limit as \( h \to 0 \) ensures we're measuring the instantaneous rate of change without being affected by the larger-scale behavior of the function. This approach allows us to precisely determine the slope of the function at a given point.

Fraction Simplification

In derivative calculations, especially those involving rational functions, simplifying fractions can drastically simplify the process. When you encounter fractions within fractions, the goal is often to combine them over a common denominator or cancel out terms to make computations cleaner and more straightforward.

In our function \( f(x) = \frac{2}{x} \), after substituting into the derivative definition, the expression \( \frac{\frac{2}{x+h} - \frac{2}{x}}{h} \) can be combined into a single fraction with a common denominator:

• \( \frac{2x - 2(x+h)}{x(x+h)} \)

From here, cancel out terms that simplify as shown:

• \( \frac{-2h}{hx(x+h)} \)

Finally, by canceling \( h \) (since the limit assumes \( h eq 0 \)), we simplify our expression into something more manageable to take the limit.

Derivative of Rational Functions

Rational functions, like \( f(x) = \frac{2}{x} \), are functions where both the numerator and denominator are polynomials. These types of functions often require careful handling when deriving them due to their potential complexity in both form and behavior. Calculating the derivative of a rational function involves employing the definition of the derivative and often requires steps like fraction simplification to solve effectively.

The process can be summarized as follows:

• Use the definition of the derivative \( f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).
• Substitute the given rational function and simplify any complex fractions that arise.
• Perform the limit calculation after simplification.

By following these steps, the derivative of \( f(x) = \frac{2}{x} \) simplifies to \( f^{\prime}(x) = \frac{-2}{x^2} \), representing how this function changes as \( x \) changes. This systematic approach to rational functions ensures that even intricate fractions are tackled efficiently and correctly, leading to accurate derivative results.