Problem 38 - Calculus an Applied Approach

Question

Find \(f^{\prime}(x)\) $$ f(x)=x^{2}+4 x+\frac{1}{x} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^{2} + 4x + 1/x\) is \(f'(x) = 2x + 4 - 1/x^{2}\).

Step 1: Identify Each Term

Here, the function \(f(x)\) consists of three separate terms: \(x^{2}\), \(4x\) and \(1/x\). Recognizing each one will ease the application of derivative rules at the next step.

Step 2: Apply Rules of Derivatives Separately To Each Term

By applying the power rule, the derivative of \(x^{2}\) is \(2x\). For \(4x\), it is just \(4\). The rule that the derivative of \(x^{-n}\) is \(-n x^{-n-1}\) implies that the derivative of \(1/x\) or \(x^{-1}\) is \(-1 * x^{-1-1} = -x^{-2}\). Hence, the derivative of \(1/x\) is \(-1/x^{2}\).

Step 3: Combine All Derivatives

Now combine all the derivatives calculated from the previous step. Hence, \(f'(x) = 2x + 4 - 1/x^{2}\).

Key Concepts

Power Rule

The power rule is a fundamental principle in derivative calculus. It is used to easily find the derivative of any term that is a power of a variable. When you have a term like \(x^n\), the power rule states that its derivative is \(nx^{n-1}\).

• For example, the derivative of \(x^2\) is \(2x^{2-1} = 2x\).
• For \(x^3\), it becomes \(3x^{3-1} = 3x^2\).

This rule simplifies the differentiation process for polynomial terms. Just multiply the term by its exponent, reduce the exponent by one, and you're done. It's especially useful in expressions where many terms involve variables raised to a power, as it allows for quick and accurate results.

Basic Derivatives

Basic derivatives refer to straightforward rules that help find the derivative of common functions. These are essential building blocks in calculus, giving you the foundation to handle more complex functions effectively.

• For constant terms like \(4\), the derivative is zero since constants don't change.
• For linear terms like \(4x\), the derivative is simply the coefficient, here \(4\).

Learning these basic derivatives makes it easier to tackle any calculus-related problem. They ensure you don't have to overthink when applying derivative calculus, letting you work with terms in their simplest forms.

Rational Functions

Rational functions consist of ratios of polynomials. They often look complex but can be understood piece by piece. In our exercise, consider the term \(\frac{1}{x}\), which is a rational function. To derive it, rewrite it using a power of \(x\), so \(\frac{1}{x} = x^{-1}\).

• Finding its derivative involves using rules we've already discussed, resulting in the derivative of \(x^{-1}\) being \(-x^{-2}\).

Handling rational functions often requires manipulation before applying the rules, especially when they appear in nonlinear forms. They illustrate how derivatives provide insights into function behavior even when the original function involves division.

Derivative Rules

Derivative rules are guidelines that make finding derivatives systematic and straightforward. They let you apply consistent methods across different kinds of functions. These rules include the product rule, quotient rule, and chain rule, among others, but the most immediate for our purposes are those for sums and powers.

• Sum Rule: The derivative of a sum is the sum of the derivatives. For a function like \(x^2 + 4x + \frac{1}{x}\), you find each term's derivative separately, then combine them.
• Power Rule: As mentioned, it is applied to each term involving powers of \(x\).

These rules apply universally, allowing solutions like \(f'(x) = 2x + 4 - \frac{1}{x^2}\) to be derived effectively. Knowing these rules lets you approach problems with confidence, knowing you're using proven mathematical principles.