Problem 44 - Brief Applied Calculus

Question

Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(x)=\frac{x^{5}+x^{3}+x}{x^{3}+x} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{2x^7 + 4x^5 + 3x^3 + x}{(x^3 + x)^2} \).

Step 1: Identify the Functions

We must first identify the numerator and the denominator for the function. Here, the function is given as \( f(x) = \frac{u(x)}{v(x)} \) where \( u(x) = x^5 + x^3 + x \) and \( v(x) = x^3 + x \).

Step 2: Quotient Rule Formula

The Quotient Rule formula for differentiation is \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \). We will use this formula to find the derivative \( f'(x) \).

Step 3: Differentiate the Numerator

Find the derivative \( u'(x) \). Since \( u(x) = x^5 + x^3 + x \), we get \( u'(x) = 5x^4 + 3x^2 + 1 \).

Step 4: Differentiate the Denominator

Find the derivative \( v'(x) \). Since \( v(x) = x^3 + x \), we get \( v'(x) = 3x^2 + 1 \).

Step 5: Apply the Quotient Rule

Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the Quotient Rule formula: \[ f'(x) = \frac{(5x^4 + 3x^2 + 1)(x^3 + x) - (x^5 + x^3 + x)(3x^2 + 1)}{(x^3 + x)^2} \].

Step 6: Simplify the Expression

Simplify the expression obtained from the previous step by expanding and combining like terms. This involves calculating the products and then subtracting them: \[ = \frac{(5x^7 + 5x^5 + 3x^5 + 3x^3 + x^3 + x) - (3x^7 + x^5 + 3x^5 + x^3)}{(x^3 + x)^2} \].

Step 7: Further Simplification

Continue to simplify by combining like terms: \[ = \frac{5x^7 + 8x^5 + 4x^3 + x - 3x^7 - 4x^5 - x^3}{(x^3 + x)^2} \].

Step 8: Final Simplified Derivative

Finish simplifying to obtain: \[ f'(x) = \frac{2x^7 + 4x^5 + 3x^3 + x}{(x^3 + x)^2} \], which is the derivative of \( f(x) \).

Key Concepts

Derivative

In calculus, the derivative of a function is a fundamental concept, representing the rate at which the function's value changes as its input changes. The derivative of a function at a particular point can be thought of as the slope of the tangent line to the graph of the function at that point. It provides crucial insights into the function's behavior, such as identifying intervals where the function is increasing or decreasing.

When working with derivatives, especially for complex functions, we often deal with more intricate rules, such as the Quotient Rule, which efficiently handles the differentiation of functions expressed as ratios of two different functions. Understanding the concept of derivatives is essential not only for theoretical mathematics but also for practical applications across science and engineering.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. For functions expressed as quotients, the differentiation process involves utilizing the Quotient Rule. This rule is particularly useful when you have a function defined as a division of two simpler functions, which we denote as \( u(x) \) and \( v(x) \).

To apply the Quotient Rule, first, determine the derivatives of both the numerator \( u'(x) \) and the denominator \( v'(x) \). With these derivatives, the Quotient Rule formula is given by:

• \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \)

This formula helps us systematically derive the derivative of the function without manually expanding every detail. By following these steps, we can efficiently manage the complexity that comes with the differentiation of quotient functions, making it a powerful tool in calculus.

Simplification of Expressions

Simplifying an expression in calculus after differentiation is an important step as it clarifies the resulting derivative expression and makes it easier to interpret or further work with. In the context of the Quotient Rule, simplification often involves several algebraic manipulations after substituting into the rule's formula.

The goal of simplification is to combine like terms and reduce the expression to its simplest form. This typically includes:

• Expanding products in the numerator.
• Combining like terms and cancelling out what you can.
• Ensuring the expression is clear and succinct, which may involve factoring when possible.

Simplification does not alter the derivative's essence but instead presents it in a form that is easier to use for further calculus operations, such as evaluating limits or finding zeros of the derivative. It makes the mathematical properties and insights more accessible, thereby enhancing both understanding and application.