Problem 31 - Brief Applied Calculus

Question

Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(x)=\frac{x^{4}+1}{x^{3}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 1 - \frac{3}{x^4} \).

Step 1: Identify the Terms

First, we identify the numerator and the denominator of the function. In the function \( f(x) = \frac{x^4 + 1}{x^3} \), the numerator \( u(x) \) is \( x^4 + 1 \) and the denominator \( v(x) \) is \( x^3 \).

Step 2: Differentiate the Numerator and Denominator

Next, we find the derivatives of the numerator and the denominator. The derivative of the numerator \( u(x) = x^4 + 1 \) is \( u'(x) = 4x^3 \). The derivative of the denominator \( v(x) = x^3 \) is \( v'(x) = 3x^2 \).

Step 3: Apply Quotient Rule

The Quotient Rule for derivatives is given by \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v u' - u v'}{v^2} \). Substitute \( u(x), v(x), u'(x), \text{ and } v'(x) \) into this formula to get the derivative of \( f(x) \).

Step 4: Substitute Values Into Quotient Rule

Substitute to find the derivative: \( f'(x) = \frac{x^3 \cdot 4x^3 - (x^4 + 1) \cdot 3x^2}{(x^3)^2} \), which simplifies to \( f'(x) = \frac{4x^6 - 3x^6 - 3x^2}{x^6} \).

Step 5: Simplify the Expression

Simplify the expression: \( f'(x) = \frac{x^6 - 3x^2}{x^6} \). Simplify further: \( f'(x) = 1 - \frac{3}{x^4} \).

Step 6: Final Answer

The derivative of the function \( f(x) = \frac{x^4 + 1}{x^3} \) is \( f'(x) = 1 - \frac{3}{x^4} \).

Key Concepts

Derivatives

The concept of derivatives is central to calculus. Derivatives measure how a function changes as its input changes. Essentially, the derivative is a way to express the slope or the rate of change of a function at a particular point. For any function \( f(x) \), its derivative, often denoted as \( f'(x) \), gives the rate at which \( f(x) \) changes with respect to \( x \). In practical terms, the derivative can tell us how fast something is moving. For instance, if \( f(x) \) represents the position of a car over time, \( f'(x) \) would represent the car's speed.There are several rules for calculating derivatives, such as the power rule, the product rule, and the quotient rule. Each rule serves different types of functions and helps find derivatives more efficiently. The quotient rule is particularly useful for functions that are expressed as one function divided by another, like the one given in our exercise.

Simplification of expressions

Simplification of expressions is the process of making algebraic expressions more manageable. It's a crucial skill in mathematics because it helps in reducing complex problems into simpler forms, which are easier to work with.When dealing with derivatives, simplification often involves combining like terms or reducing fractions. For the function \( f(x) = \frac{x^4 + 1}{x^3} \), once we apply the quotient rule, we get a complex fraction. Simplification is used to make this expression easier to read and interpret.During simplification:

• Look for common factors in the numerator and the denominator.
• Cancel out terms that appear in both the top and bottom of the fraction.
• Always aim to express the final result in the simplest terms possible, as this often reveals insights into the behavior of the function.

Using these strategies ensures that solutions are not only accurate but also clear and concise, reflecting the derivative's impact more transparently.

Function differentiation

Function differentiation is a fundamental process in calculus used to find the derivative of a given function. Differentiation involves applying specific rules to derive formulas that describe the rate of change of a function.To differentiate a function that is a quotient of two functions, like \( f(x) = \frac{x^4 + 1}{x^3} \), we use the quotient rule. This rule states that the derivative of \( \frac{u}{v} \) is \( \frac{v u' - u v'}{v^2} \), where \( u \) and \( v \) are the numerator and denominator, respectively.Steps in function differentiation using the quotient rule include:

• Identifying the numerator and denominator, \( u \) and \( v \).
• Finding their respective derivatives, \( u' \) and \( v' \).
• Applying the quotient rule formula: replace and simplify.

This process is systematic and provides a framework for calculating derivatives of more complex rational functions. By following this approach, we achieve an accurate description of how functions behave and change.