Problem 48 - Calculus and its applications

Question

Find \(y^{\prime}\) $$ y=\frac{2}{x}-\frac{x}{2} $$

Step-by-Step Solution

Answer

\( y' = -\frac{2}{x^2} - \frac{1}{2} \)

Step 1: Differentiate the First Term

The first term of the function is \( \frac{2}{x} \), which can be rewritten as \( 2x^{-1} \) for differentiation purposes. Using the power rule, \( \frac{d}{dx}(x^n) = nx^{n-1} \), the derivative of \( 2x^{-1} \) is \( -2x^{-2} \).

Step 2: Differentiate the Second Term

The second term of the function is \( -\frac{x}{2} \). Applying the constant rule and the fact that the derivative of \( x \) is 1, we get \( \frac{d}{dx}\left(-\frac{x}{2}\right) = -\frac{1}{2} \).

Step 3: Combine the Derivatives

Combine the derivatives obtained in Steps 1 and 2 to find \( y' \). Thus, \( y' = -2x^{-2} - \frac{1}{2} \).

Step 4: Write the Answer in Fraction Form

Express \( -2x^{-2} \) as \( -\frac{2}{x^2} \). Thus, the final form of the derivative is: \[ y' = -\frac{2}{x^2} - \frac{1}{2} \]

Key Concepts

Power Rule

The Power Rule is a fundamental tool in calculus for finding the derivative of a function that is in the form of a power. Whenever you have a term like \(x^n\), the Power Rule comes into play. It tells us that the derivative of \(x^n\) with respect to \(x\) is \(n \cdot x^{n-1}\). This means you bring down the exponent as a multiplier and decrease the exponent by one.

In our original exercise, we started with the term \(2x^{-1}\), which initially looks complex with the negative exponent. However, by applying the Power Rule, the differentiation becomes straightforward:
\(-1\) (the exponent) \,\cdot\, 2 (constant) = \(-2\)
\(x^{-1-1} = x^{-2}\)
Thus, the derivative of \(2x^{-1}\) is \(-2x^{-2}\). This showcases the efficiency of the Power Rule in simplifying differentiation.
Understanding and practicing this rule will significantly enhance your problem-solving capabilities in calculus.

Constant Rule

The Constant Rule is another essential rule in calculus which states that the derivative of a constant multiplied by a function is just the constant multiplied by the derivative of the function. In simpler terms, if you have a term where a constant is multiplied by a variable (like \(-\frac{x}{2}\)), you can pull the constant out and then differentiate the remaining variable separately.

Let's apply the Constant Rule to our function. The term \(-\frac{x}{2}\) can be seen as \(-\frac{1}{2} \cdot x\). According to the Constant Rule, it becomes:

• The constant \(-\frac{1}{2}\) stays the same.
• The derivative of \(x\) is simply \(1\).

Therefore, differentiating \(-\frac{x}{2}\) gives us \(-\frac{1}{2} \cdot 1 = -\frac{1}{2}\). This rule simplifies many derivative calculations by allowing you to focus on one part of the term at a time.

Differentiation Steps

Differentiating a function with multiple terms involves handling each term independently before combining the results. This process is crucial for solving derivatives accurately.

Here's how we applied these steps:

• **Step 1:** For the term \(\frac{2}{x}\), write it as \(2x^{-1}\). Apply the Power Rule to find its derivative, which became \(-2x^{-2}\).
• **Step 2:** For the term \(-\frac{x}{2}\), apply the Constant Rule and find that its derivative is \(-\frac{1}{2}\).
• **Step 3:** After finding the derivatives of each term, combine these results to get the overall derivative of the function.
• **Step 4:** Simplify the answer if possible, such as converting \(-2x^{-2}\) into \(-\frac{2}{x^2}\).

By following these systematic differentiation steps, you ensure no part of the function is overlooked, providing a clear path from a complex expression to an understandable derivative.