Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

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. $$ f(x)=\frac{x+2}{x} $$

Step-by-Step Solution

Answer

The derivative of \(f\) is \(f'(x)=-2/x^2\).

Step 1: Simplify the Expression

Divide each term in the numerator by \(x\). This gives \(f(x)=\frac{x}{x}+\frac{2}{x}=1+\frac{2}{x}=1+2x^{-1}\)

Step 2: Apply the Power Rule

The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Apply this rule to each term to find the derivative: \(f'(x)= 0 + (-1)*2x^{-1-1}= 0 - 2x^{-2}\).

Step 3: Simplify the Result

Express \(f'(x)\) in terms of positive exponents: \(f'(x)= -2x^{-2} = -2/x^2\).

Key Concepts

Power Rule Differentiation

When confronted with the task of finding a function's derivative, the power rule is an essential tool. This rule simplifies the process of differentiation by providing an easy formula when dealing with powers of variables. The basic principle behind power rule differentiation states that for any function in the form of \( x^n \), where \( n \) is a real number, the derivative is \( nx^{n-1} \).

So, if you have a function like \( f(x) = x^3 \), the derivative, using the power rule, would be \( f'(x) = 3x^{2} \). This same principle applies to negative exponents; for example, if you have the term \( 2x^{-1} \), its derivative would be \( -2x^{-2} \). This simplicity allows students to tackle calculus homework problems with confidence and efficiency. Additionally, the power rule helps in avoiding more complex rules like the Quotient Rule, if the given function can first be simplified.

Simplifying Algebraic Expressions

Simplification in calculus is a powerful technique for making complex problems more manageable. Algebraic simplification often makes differentiation easier and can result in clearer, more concise solutions. In calculus homework problems, such as finding the derivative of a function presented as a quotient of polynomials, students should always consider simplification as a first step.

For example, the expression \(\frac{x+2}{x}\) can be simplified by dividing each term in the numerator by the denominator, turning it into \(1+2x^{-1}\). In doing so, the derivative becomes much simpler to compute, bypassing the need for the Quotient Rule. Simplifying algebraic expressions is not just about making the equation look neater; it is about breaking down the equation into simpler components to make the subsequent calculus more straightforward.

Calculus Homework Problems

Calculus students often face a variety of homework problems that test their understanding of differentiation, integration, and the application of various mathematical rules. The goal of these exercises is to strengthen problem-solving skills and apply theoretical knowledge to practical situations.

When tackling homework problems, especially those involving derivatives, it is instrumental to:

• Review and understand relevant differentiation rules, such as the power rule.
• Simplify expressions before attempting to differentiate.
• Practice regularly to become familiar with different types of functions and their derivatives.

Always remember, the aim is not just to find the right answer, but also to understand the underlying processes that lead to the solution. This understanding will enable students to approach more complex calculus problems with ease and result in a more robust grasp of the subject matter.