Problem 21 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{1}{6} x^{3}+\frac{1}{2} x^{2}+x+1 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are asked to find the derivative of the function \( f(x) = \frac{1}{6} x^{3} + \frac{1}{2} x^{2} + x + 1 \). This involves applying the rules of differentiation to each term of the polynomial.

Step 2: Differentiate Term by Term

Differentiate each term of the function separately by applying the power rule, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \).

Step 3: Differentiate the First Term

Using the power rule, differentiate \( \frac{1}{6}x^3 \):\[ \frac{d}{dx} \left( \frac{1}{6}x^3 \right) = \frac{1}{6} \cdot 3x^{3-1} = \frac{1}{2}x^2 \]

Step 4: Differentiate the Second Term

Differentiate \( \frac{1}{2}x^2 \) using the power rule:\[ \frac{d}{dx} \left( \frac{1}{2}x^2 \right) = \frac{1}{2} \cdot 2x^{2-1} = x \]

Step 5: Differentiate the Third Term

The derivative of \( x \) with respect to \( x \) is 1, since \( \frac{d}{dx}[x] = 1 \).

Step 6: Differentiate the Constant Term

The derivative of a constant (1 in this case) is 0. This is because the rate of change of any constant value is zero.

Step 7: Compile the Derivatives

Combine all the derivatives to form the derivative of the entire function. \[ f'(x) = \frac{1}{2}x^2 + x + 1 + 0 = \frac{1}{2}x^2 + x + 1 \]

Key Concepts

Understanding the Power Rule

The power rule is a fundamental principle in calculus and is incredibly useful when finding derivatives of polynomial functions. This rule is simple and states that if you have a function of the form \( x^n \), the derivative is \( nx^{n-1} \). Let's break this down:

• "n" represents the exponent of the term.
• The derivative turns the exponent into a coefficient (\( n \)).
• The new exponent is one less than the original (\( n-1 \)).

For example, if you have \( 7x^4 \), using the power rule:

• First, multiply the exponent (4) by the coefficient (7), giving you 28.
• Then, reduce the exponent by 1, changing \( x^4 \) to \( x^3 \).
• The derivative is \( 28x^3 \).

This rule makes it straightforward to differentiate any polynomial term. It's one of the first rules taught in calculus because it applies to any power of \( x \), thus making it an essential tool for students.

Differentiation Basics

Differentiation is the process of finding a derivative, which is essentially the rate of change of a function. It's a core concept in calculus and is used across many applications in mathematics and science. Here’s how differentiation works:

• It provides the slope of the tangent line at any given point on a curve.
• For a function \( f(x) \), its derivative \( f'(x) \) is found using certain rules such as the power rule, product rule, and quotient rule, depending on the complexity of the function.
• Each term of a polynomial is differentiated separately then combined to form the derivative of the entire function.

For a polynomial like \( f(x) = \frac{1}{6}x^3 + \frac{1}{2}x^2 + x + 1 \):

• You differentiate \( \frac{1}{6}x^3 \) to get \( \frac{1}{2}x^2 \).
• Differentiate \( \frac{1}{2}x^2 \) to find \( x \).
• The term \( x \) becomes 1 after differentiation.
• The constant \( 1 \) becomes 0, because the derivative of a constant is always 0.

Differentiation simplifies the process of finding how a function changes, providing insights into the behavior of functions.

Connecting to Calculus

Calculus is the branch of mathematics that studies continuous change, and differentiation is one of its two major operations, along with integration. Here are some reasons why calculus, specifically differentiation, is invaluable:

• It allows us to calculate the rate at which one quantity changes with respect to another. This forms the basis for topics such as velocity, acceleration, and optimization problems.
• In scientific and engineering fields, calculus helps model real-world systems to predict their behavior, such as how a car's speed changes over time.
• Differentiation is used to find the minima and maxima of functions which are crucial in fields like economics to maximize profits or minimize costs.

By using calculus, we can understand and describe the motion and change that occur in natural phenomena. The derivative, as taught through differentiation, serves as a foundational concept, without which many advanced mathematical topics would not be possible.