Problem 48 - Brief Applied Calculus

Question

Differentiate each function by rewriting before and after differentiating, as on page 130 . $$ y=\frac{x^{2}}{4} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{2}x \).

Step 1: Rewrite the Function

Rewrite the function from a fraction to a form more suitable for differentiation. The function is given as \( y = \frac{x^2}{4} \). This can be rewritten as \( y = \frac{1}{4} x^2 \). By doing so, the expression becomes easier to handle when applying differentiation rules.

Step 2: Differentiate the Function

Use the power rule to differentiate the rewritten function. The power rule states that \( \frac{d}{dx} [ax^n] = anx^{n-1} \). Applying this to \( y = \frac{1}{4} x^2 \), the derivative \( \frac{dy}{dx} \) is calculated as follows: \( \frac{dy}{dx} = \frac{1}{4} \, (2x^{2-1}) = \frac{1}{4} \, (2x) = \frac{2}{4}x = \frac{1}{2}x \).

Key Concepts

Power Rule

The power rule is a simple and widely used technique in calculus for differentiating functions of the form \( ax^n \). Understanding and applying the power rule is essential for efficient differentiation. This rule states that the derivative of \( ax^n \) is \( anx^{n-1} \). In simpler terms, you multiply the coefficient \( a \) by the exponent \( n \), and then subtract 1 from the exponent.

Here is a quick breakdown:

• Identify the base and exponent in the term (e.g., in \( x^2 \), the base is \( x \), and the exponent is 2).
• Multiply the coefficient (the number in front of the \( x \), like \( a \)) by the exponent.
• Reduce the exponent by 1.
• The new expression \( anx^{n-1} \) is your derivative.

So, for our example \( y = \frac{1}{4}x^2 \), applying the power rule, we take the 2 (from \( x^2 \)) and multiply it with \( \frac{1}{4} \), giving us \( \frac{1}{2}x^{2-1} \), or simply \( \frac{1}{2}x \).

This straightforward process is why the power rule is so powerful and often one of the first differentiation techniques covered in calculus.

Rewriting Functions

Before differentiating, sometimes it's necessary to rewrite a function into a more manageable form. This approach allows for easier application of differentiation rules such as the power rule.

Consider the original function \( y = \frac{x^2}{4} \). At a glance, it seems tricky to differentiate due to its fractional form. By rewriting it as \( y = \frac{1}{4}x^2 \), we make the function more straightforward to handle. Converting a complex function to a polynomial form allows us to utilize basic rules of calculus without extra hassle.

Steps for rewriting functions include:

• Look for terms that can become coefficients, as seen with \( \frac{1}{4} \) in front of \( x^2 \).
• Translate the function into a sum of terms, if necessary.

Rewriting changes the expression's appearance but not its value, making differentiation simpler.

Differentiation Techniques

Differentiation is the process of finding a derivative, a fundamental concept in calculus that helps determine the rate of change. Various techniques exist, each suited for different types of functions, and mastering them allows you to tackle a wide range of problems.

The key differentiation techniques include:

• The Power Rule: Efficiently applied to terms with \( x^n \), as demonstrated earlier.
• Product and Quotient Rules: For differentiating products or quotients of functions, these rules become essential.
• Chain Rule: Used when dealing with composite functions, ensuring each part is appropriately differentiated.

In our problem, only the power rule was required since the function \( y = \frac{1}{4}x^2 \) was in a suitable form for this straightforward method. By knowing and applying the right technique, differentiation becomes a systematic process leading directly to the solution.