Problem 6 - The Calculus with Analytic Geometry

Question

f(x)=2 x+\frac{1}{2 x}

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = 2x + \frac{1}{2x} \) is \( f'(x) = 2 - \frac{1}{2x^2} \).

Step 1: Identify the Function

The function given is a combination of a linear term and a reciprocal term: \[ f(x) = 2x + \frac{1}{2x} \]

Step 2: Find the Derivative

Use differentiation rules. The derivative of the linear term, \(2x\), is \(2\). The derivative of the reciprocal term, \( \frac{1}{2x} \), is \( \frac{-1}{2x^2} \): \[ f'(x) = 2 - \frac{1}{2x^2} \]

Step 3: Simplify the Derivative

Combine the terms to get the simplified form of the derivative: \[ f'(x) = 2 - \frac{1}{2x^2} \]

Key Concepts

Derivative Rules

Derivatives are a foundational concept in calculus. They help us understand how a function changes as its input changes. When dealing with derivatives, several rules are essential. The most used are the power rule, the sum rule, and the reciprocal rule.

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). For example, the derivative of \(x^2\) is \(2x\). The sum rule indicates that the derivative of a sum of functions is the sum of their derivatives. If you have \(f(x) = g(x) + h(x)\), then \(f'(x) = g'(x) + h'(x)\). Lastly, the reciprocal rule is a bit more complicated; it states that the derivative of \((\frac{1}{x})\) is \(-\frac{1}{x^2}\).

Using these rules simplifies complex differentiation tasks. In our example, we deal with both a linear term and a reciprocal term. By using the sum rule, we split the function and apply the power and reciprocal rules individually.

Function Analysis

Function analysis involves understanding the behavior and properties of functions. This can include identifying key features like domain, range, intercepts, and asymptotes.

For our function, \(f(x) = 2x + \frac{1}{2x}\), we can break it down:

• The domain is all real numbers excluding \(x = 0\), as you cannot divide by zero.
• The linear term, \(2x\), grows indefinitely as \(x\) increases or decreases.
• The reciprocal term, \(\frac{1}{2x}\), approaches zero as \(x\) moves away from zero, but spikes towards infinity as \(x\) gets closer to zero.

Combining these, we can see the function's behavior varies significantly for different values of \(x\). It's also useful to look at the function's limits and study how it approaches different values to understand its overall behavior.

Reciprocal Function

A reciprocal function is one where the variable is in the denominator, like \(\frac{1}{x}\). Reciprocal functions have unique characteristics:

• They are undefined at \x = 0\.
• As \x\ gets very large (positive or negative), the reciprocal gets very small, approaching zero but never actually reaching it.
• As \x\ approaches zero, the function's value grows towards infinity.

In the given function \(f(x) = 2x + \frac{1}{2x}\), the term \(\frac{1}{2x}\) is the reciprocal. Even though it's part of a larger function, its characteristics influence the function's overall behavior. For instance, \(\frac{1}{2x}\) approaches zero as \x\ moves away from zero, but it introduces an asymptotic behavior near zero.

Understanding reciprocal functions is essential in analyzing overall functions, especially when they appear with other terms in composite functions.