Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 8 , find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=3 x^{2}+3 x+3+3 x^{-1}+3 x^{-2} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=3 x^{2}+3 x+3+3 x^{-1}+3 x^{-2}\) is \(f'(x) = 6x + 3 -3x^{-2} + 6x^{-3}\).

Step 1: Identify Each Term

The given function \(f(x)=3 x^{2}+3 x+3+3 x^{-1}+3 x^{-2}\) contains five different terms which are \(3 x^{2}\), \(3x\), \(3\), \(3 x^{-1}\) and \(3 x^{-2}\). Each of these terms will be differentiated separately using the power rule.

Step 2: Differentiate Each Term Using the Power Rule

Applying the power rule, the derivative of the first term \(3x^{2}\) is \(6x\), the derivative of the second term \(3x\) is \(3\), the derivative of the third term \(3\) is \(0\), the derivative of the fourth term \(3x^{-1}\) is \(-3x^{-2}\), and the derivative of the fifth term \(3x^{-2}\) is \(6x^{-3}\).

Step 3: Combine to Form the Complete Derivative

Combine the results from the previous step to get the derivative of the whole function. Thus, \(f'(x) = 6x + 3 -3x^{-2} + 6x^{-3}\).

Key Concepts

Calculus

Calculus is a fascinating and essential branch of mathematics that deals with the study of change. It helps us understand the behavior of functions and how they alter over time or in response to different variables. There are two main fields within calculus: differential and integral calculus. Differential calculus, which is the focus here, concerns itself with the concept of the derivative, an instrument for examining rates at which quantities change.

In the student's exercise, we encounter a function needing differentiation to find the rate at which it changes. Calculus provides tools like the power rule to handle these problems efficiently, revealing how calculus is not just theoretical, but practically applicable in analyzing and solving real-world problems.

Derivative

A derivative represents the rate at which a function is changing at any given point and is a fundamental concept in calculus. For a function of one variable, such as in the exercise, the derivative at any point gives the slope of the tangent line to the function's graph at that point. This slope is an indication of how steep the graph is, and thus, how quickly the function value is changing.

When differentiating a function such as the one in the exercise, we apply rules from calculus to obtain the function's derivative, which is denoted as \( f'(x) \) or \( \frac{df}{dx} \). The process of finding the derivative is often referred to as differentiation.

Power Rule

The power rule is a fundamental technique in differential calculus for taking derivatives of power functions. It states that if you have a function \( f(x) = x^n \), where \( n \) is any real number, the derivative of that function is \( f'(x) = nx^{n-1} \). This is a fast and powerful method to differentiate functions term by term when they are in the form of a power function.

For example, given the term \( 3x^2 \) in the student's exercise, we apply the power rule by multiplying the exponent by the coefficient and then decrementing the exponent by one. The differentiation of \( 3x^2 \) results in \( 6x \), which is the first term in the complete derivative of the original function.

Term by Term Differentiation

Term by term differentiation is a practical approach when a function is composed of several terms added or subtracted together, as is the case with the student's exercise. This technique allows us to differentiate each term independently and then recombine them for the final derivative of the entire function.

Starting with the function \( f(x) = 3x^{2} + 3x + 3 + 3x^{-1} + 3x^{-2} \), we break it down into individual terms and apply the differentiation rules such as the power rule for each one. After finding the derivatives of all terms, they are added (or subtracted, as necessary) to provide us with the comprehensive derivative \( f'(x) = 6x + 3 - 3x^{-2} + 6x^{-3} \), effectively showcasing how term by term differentiation streamlines the process of finding a function's overall rate of change.