Problem 4 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=\frac{1}{2}\left(x^{4}+7\right)$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = 2x^3\)

Step 1: Rewriting the Function

Write the function in terms of its components:\[y = \frac{1}{2} \left( x^4 + 7 \right)\] This can be expressed as:\[y = \frac{1}{2}x^4 + \frac{1}{2} \times 7\] which simplifies to:\[y = \frac{1}{2}x^4 + \frac{7}{2}.\]

Step 2: Differentiate Each Term Separately

Differentiate each part of the expression:- The term \(\frac{1}{2}x^4\) becomes \(2x^3\) after differentiation, because:\[\frac{d}{dx} \left( \frac{1}{2}x^4 \right) = \frac{1}{2} \times 4x^3 = 2x^3.\] - The term \(\frac{7}{2}\) is a constant, so its derivative is 0.

Step 3: Combine the Results

Combine the derivatives obtained:\[\frac{dy}{dx} = 2x^3 + 0 = 2x^3.\] So, the derivative of the function is \(2x^3\).

Key Concepts

Understanding Differentiation

Differentiation is a fundamental concept in calculus, focusing on how a function changes at any given point. It allows us to calculate the derivative, which represents the rate of change or the slope of the function at a particular location. In simpler terms, if you think of a curve as a series of tiny steps, differentiation tells you how steeply each step tilts. This can be crucial in many fields like physics, engineering, and economics where understanding how things change is essential.

• Derivatives give us important information about the behavior of functions, such as finding where they increase or decrease.
• They help in analyzing real-world phenomena mathematically.

Differentiation is used to solve various problems, including finding the maximum or minimum points of a function, which is key in optimizing outcomes in real-world applications. Different techniques exist for differentiating various types of functions, and mastering these techniques opens the door to solving complex problems.

Applying the Power Rule

The power rule is a handy tool when differentiating algebraic functions of the form \(x^n\), where \(n\) is any real number. It's a simple yet powerful rule that makes differentiation straightforward. The power rule states that to differentiate \(x^n\), you multiply by the exponent \(n\) and then subtract one from the exponent. Using this, \(\frac{d}{dx}(x^n) = nx^{n-1}\).
For instance, to differentiate \(x^4\), apply the power rule as follows:

• Multiply by the exponent: \(4 \times x^{4-1} = 4x^3\).

In our exercise, we applied the power rule to the term \(\frac{1}{2}x^4\), which became \(2x^3\) upon differentiation. The power rule is crucial because it simplifies the process of finding derivatives for polynomial functions, making it easier to tackle calculus problems without complex effort.

Solving Calculus Problems

Calculus problems often involve complex functions and the application of various rules and techniques like differentiation. The goal can be as diverse as finding rates of change, determining the shape of a curve, or optimizing some quantity. In solving these problems, recognizing the type of function you are working with is critical.
Start by identifying all parts of the function and see which rules apply. For polynomial functions, the power rule is typically the first step. As demonstrated in the exercise, it's important to rewrite the function if necessary to isolate terms that are easy to differentiate.

• Rewriting: Breaking down the function into simple parts.
• Applying Rules: Using differentiation techniques such as the power rule.
• Combining Results: Ensure you combine derivatives correctly to get the final answer.

Puzzling through calculus problems can be intimidating at first, but by taking it step by step and applying the right techniques, you can solve them with confidence. Practice is key. With steady effort, solving calculus problems becomes an exciting challenge rather than a daunting task.