Problem 15 - Calculus and its applications

Question

Differentiate two ways: first, by using the Quotient Rule; then, by dividing the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(g(x)=\frac{3 x^{7}-x^{3}}{x}\)

Step-by-Step Solution

Answer

Both methods yield the same derivative: \( g'(x) = 18x^5 - 2x \).

Step 1: Differentiate using Quotient Rule

The Quotient Rule is used to differentiate a function of the form \( \frac{u}{v} \), where \( u = 3x^7 - x^3 \) and \( v = x \). The rule states that:\( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \)First, find the derivatives:\( u' = 21x^6 - 3x^2 \)\( v' = 1 \)Now apply the Quotient Rule:\( g'(x) = \frac{x(21x^6 - 3x^2) - (3x^7 - x^3) \cdot 1}{x^2} \)Simplify the expression:\( g'(x) = \frac{21x^7 - 3x^3 - 3x^7 + x^3}{x^2} \)Combine like terms:\( g'(x) = \frac{18x^7 - 2x^3}{x^2} \)Simplify further to get:\( g'(x) = 18x^5 - 2x \).

Step 2: Simplify the expression before differentiating

Before differentiating, simplify \( g(x) = \frac{3x^7 - x^3}{x} \) by dividing each term by \( x \):\( g(x) = \frac{3x^7}{x} - \frac{x^3}{x} \)\( g(x) = 3x^6 - x^2 \)Now differentiate directly:\( g'(x) = \frac{d}{dx}(3x^6) - \frac{d}{dx}(x^2)\)Calculate each derivative:\( \frac{d}{dx}(3x^6) = 18x^5 \)\( \frac{d}{dx}(x^2) = 2x \)Thus, \( g'(x) = 18x^5 - 2x \).

Step 3: Comparison and verification using a graphing tool

Both methods result in \( g'(x) = 18x^5 - 2x \). Use a graphing calculator to plot \( g(x) = \frac{3x^7 - x^3}{x} \) and its derivative \( g'(x) = 18x^5 - 2x \). Verify that the derivative function matches the slope of the original function's curve at various points.

Key Concepts

Quotient Rule

The Quotient Rule is a key principle in calculus for differentiating functions that are expressed as a ratio of two other functions. When you encounter a function of the form \( \frac{u}{v} \), where both \( u \) and \( v \) are differentiable functions, the Quotient Rule provides a structured method to find the derivative. The rule is expressed as follows:\[\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}\]This rule states that the derivative of \( \frac{u}{v} \) is computed by:

• Multiplying \( v \) by the derivative of \( u \), \( u' \).
• Subtracting the product of \( u \) and the derivative of \( v \), \( v' \).
• Finally, dividing the result by \( v^2 \), the square of \( v \).

Use this rule whenever you need to differentiate a quotient, ensuring that you carefully follow each step for accurate results.

Simplifying Expressions

Simplifying expressions before differentiating can save you time and effort, especially with functions that are composed of multiple terms. In our original exercise, the function \( g(x) = \frac{3 x^{7}-x^{3}}{x} \) was simplified by dividing each term in the numerator by \( x \), resulting in a simpler expression:\[g(x) = 3x^6 - x^2\]Simplifying expressions is a useful strategy when:

• The numerator can be easily divided by the denominator, reducing the function to its simplest form.
• The simplification leads to expressions that are straightforward to differentiate directly.
• You want to avoid complex or lengthy quotient rule calculations when a simpler path is available.

After simplification, differentiation is straightforward because each term is a basic power of \( x \):

• \( \frac{d}{dx}(3x^6) = 18x^5 \)
• \( \frac{d}{dx}(x^2) = 2x \)

This method could help you avoid potential algebraic mishaps encountered with larger expressions.

Graphing Calculator

Graphing calculators are powerful tools that allow you to visualize functions and their derivatives. In the context of this exercise, a graphing calculator can help you confirm the accuracy of your computed derivative \( g'(x) = 18x^5 - 2x \).Here's how to use a graphing calculator for the verification:

• First, input the original function \( g(x) = \frac{3x^7 - x^3}{x} \) into the calculator.
• Next, enter the derivative \( g'(x) = 18x^5 - 2x \) for graphing.
• Examine both graphs: the original function and its derivative.
• Check that the slope of the original function's graph aligns with the derivative function at various points.

A graphing calculator, therefore, serves as a visual aid to confirm that your derivative calculations are correct, making it a useful tool in your calculus learning toolkit.