Problem 110 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=x^{4}+\frac{2}{x}$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 4x^3 - 2x^{-2} \).

Step 1: Identify the functions to differentiate

The given function is \( f(x) = x^4 + \frac{2}{x} \). This function is the sum of two simpler functions: \( x^4 \) and \( \frac{2}{x} \).

Step 2: Differentiate the first term

Apply the power rule, which states that the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \). For the term \( x^4 \), \( n = 4 \), so the derivative is \( 4x^3 \).

Step 3: Differentiate the second term

Rewrite \( \frac{2}{x} \) as \( 2x^{-1} \). Use the power rule again. The derivative of \( 2x^{-1} \) is \(-2x^{-2} \), because \( 2(-1)x^{-1-1} = -2x^{-2} \).

Step 4: Sum the derivatives

Combine the derivatives from Step 2 and Step 3 to find \( f'(x) \). Thus, \( f'(x) = 4x^3 - 2x^{-2} \).

Key Concepts

Power Rule

The power rule is one of the simplest and most frequently used differentiation rules in calculus. It allows you to quickly find the derivative of functions where a term is raised to a power. All you need to remember is: if you have a term of the form \(x^n\), its derivative is \(nx^{n-1}\). This rule is essential because it enables swift computation of derivatives without delving into the complexities of limits. The power rule simplifies the differentiation process, especially when dealing with polynomials.

• For a term \(x^4\), use the power rule to get \(4x^3\).
• If the term is \(x^{-1}\) (which is what \(\frac{1}{x}\) becomes when rewritten), its derivative is \(-x^{-2}\).

Always reduce the exponent by 1 and multiply by the original exponent. This straightforward method will serve you well in most calculus problems that involve polynomial terms.

Differentiation

Differentiation is a core concept in calculus that deals with finding the rate at which a function is changing at any given point. It is a fundamental tool used to explore how variables interact and change over time or through a range of inputs.
In practical terms, differentiation gives us the slope or gradient of the tangent line to a curve at any point. This slope tells us how fast or slow a function is increasing or decreasing.

• To differentiate, first accurately identify each term within the function.
• Apply appropriate differentiation rules, like the power rule, to each term separately.
• Combine the derived terms to find the overall derivative of the function.

Differentiation helps us understand not just static values, but dynamic change - whether in math, physics, or any other field that involves modeling with equations.

Calculus Problems

Calculus problems often involve applying various rules and principles to find derivatives and integrals. These problems can range from finding the slope of a line tangent to a curve to solving complex real-world phenomena that require precise modeling.
Each calculus problem can be approached systematically by:

• Breaking down complicated functions into simpler parts.
• Applying known differentiation rules, like the power rule, to each part.
• Recombining the results to obtain a final answer.

It's crucial to thoroughly understand the underlying principles, such as differentiation rules, so you can adapt to different kinds of problems.
Solving these problems helps in developing critical thinking and analytical skills that can be applied across various scientific disciplines.