Problem 9 - Calculus

Question

Find the derivatives of the functions in exercise. Even if \(n\) is negative or a fraction, the derivative of \(x^{n}\) is \(n x^{n-1}\). $$ 1+x+\frac{1}{2} x^{2}+\frac{1}{6} x^{3}+\frac{1}{24} x^{4} $$

Step-by-Step Solution

Answer

The derivative is \(1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3\).

Step 1: Identify Each Term's Power of x

Firstly, we need to identify and clearly note the power of each term. In the given function:- The first term is constant, 1.- The second term, coefficient of the first-order, is \(x\).- The third term is \(\frac{1}{2}x^2\).- The fourth term is \(\frac{1}{6}x^3\).- The fifth term is \(\frac{1}{24}x^4\).

Step 2: Apply the Power Rule to Each Term

Use the derivative formula \(\frac{d}{dx}(x^n) = nx^{n-1}\) for each non-constant term:- Constant: The derivative of 1 is 0.- For \(x\), the derivative is \(1 \times x^{1-1} = 1\).- For \(\frac{1}{2}x^2\), the derivative is \(2 \times \frac{1}{2}x^{2-1} = x\).- For \(\frac{1}{6}x^3\), the derivative is \(3 \times \frac{1}{6}x^{3-1} = \frac{1}{2}x^2\).- For \(\frac{1}{24}x^4\), the derivative is \(4 \times \frac{1}{24}x^{4-1} = \frac{1}{6}x^3\).

Step 3: Combine the Derivatives

Write down the combined derivatives of each term into a single expression:\[0 + 1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3\]So, the derivative of the entire function is \(1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3\).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in calculus for finding derivatives of polynomial expressions. It's a straightforward technique that allows you to determine the rate of change of a function. When using the power rule, each term's exponent is important. If you have a term like \(x^n\), the power rule states that its derivative is \(n \times x^{n-1}\). To apply the power rule:

• Identify the power of each term (the exponent \(n\)).
• Multiply the entire term by the exponent \(n\).
• Subtract one from the original exponent \(n\) to get the new exponent of \(x\).

This rule simplifies finding derivatives, as you can see with expressions such as \(x^2\), \(x^3\), and higher-order powers. Remember, constants (like the number 1) simply vanish in differentiation, as they do not depend on \(x\) and their rate of change is zero. The power rule is vital for calculus basics and will recur frequently when solving derivatives.

Delving into Polynomial Functions

Polynomial functions are expressions composed of variables raised to various powers, with each term's power being a non-negative integer. They can take many forms, from simple quadratic functions to complex expressions involving multiple terms. In calculus, polynomial functions are essential for modeling real-world scenarios, ranging from physics to economics.Each term in a polynomial function has a coefficient that multiplies the base \(x\) raised to an exponent. For instance, the expression \(\frac{1}{2}x^2\) represents a polynomial term with the base \(x\), an exponent of 2, and a coefficient of \(\frac{1}{2}\).To find derivatives of polynomial functions:

• Apply the power rule to each term.
• Simplify the expression by combining results.

The beauty of polynomial functions lies in their simplicity and structure, allowing straightforward application of calculus techniques like differentiation. They're foundational to understanding more complex functions in higher calculus.

Exploring Calculus Basics

Calculus is the mathematical study of change, divided mainly into differential and integral calculus. One of its core tasks involves differentiation, the process of finding the derivative of a function. This tells us how the function changes, moment by moment. Basics covered in differential calculus include:

• Differentiation of simple and complex functions using rules like the power rule.
• Understanding constant functions (derivative zero) and linear functions.
• Recognizing and differentiating polynomial functions.

Derivatives are powerful tools that help in understanding the behavior of functions, slopes of curves, and make predictions about future values of functions. They form the foundation for more advanced topics like finding maxima and minima, optimization problems, and studying the motion of particles. By getting comfortable with these calculus basics, you build a strong foundation, enabling you to tackle more complex mathematical challenges with confidence.