Problem 16 - 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. \(f(x)=\frac{2 x^{5}+x^{2}}{x}\)

Step-by-Step Solution

Answer

The derivative \( f'(x) = 8x^3 + 1 \) obtained through both methods is consistent. Verified by a graphing calculator.

Step 1: Identify the Function and Set Up for Quotient Rule

The function given is \( f(x) = \frac{2x^5 + x^2}{x} \). We will first differentiate using the Quotient Rule. The Quotient Rule states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \), where \( u(x) = 2x^5 + x^2 \) and \( v(x) = x \).

Step 2: Differentiate Using the Quotient Rule

Calculate the derivatives of \( u(x) \) and \( v(x) \):- \( u'(x) = \frac{d}{dx}(2x^5 + x^2) = 10x^4 + 2x \).- \( v'(x) = \frac{d}{dx}(x) = 1 \).Now apply the Quotient Rule: \[ f'(x) = \frac{(10x^4 + 2x)x - (2x^5 + x^2) \cdot 1}{x^2} \] Simplifying the expression:\[ f'(x) = \frac{10x^5 + 2x^2 - 2x^5 - x^2}{x^2} \] => \[ f'(x) = \frac{8x^5 + x^2}{x^2} \] => \[ f'(x) = 8x^3 + 1 \].

Step 3: Simplify Before Differentiating

Simplify \( f(x) \) by dividing each term of the numerator by the denominator:\[ f(x) = \frac{2x^5}{x} + \frac{x^2}{x} = 2x^4 + x \].Now, differentiate the simplified function:- \( f'(x) = \frac{d}{dx}(2x^4 + x) = 8x^3 + 1 \).

Step 4: Compare the Results

Both methods yielded the same derivative result:- From the Quotient Rule: \( f'(x) = 8x^3 + 1 \).- From Simplification: \( f'(x) = 8x^3 + 1 \).This consistency confirms that the calculations are correct.

Step 5: Verify with a Graphing Calculator

Using a graphing calculator, input the original function and confirm that the calculated derivative \( f'(x) = 8x^3 + 1 \) matches the derivative feature of the calculator. The graph should illustrate the slope values at various points according to the derived function.

Key Concepts

Differentiation

Differentiation is the process used to compute the derivative of a function. It helps us find how a function changes at any given point. When we differentiate using the Quotient Rule, we apply a specific formula designed for dividing two functions. The rule states that if you have a function in the form of a quotient, like \( f(x) = \frac{u(x)}{v(x)} \), the derivative \( f'(x) \) can be found using:

• \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \)

This rule is essential when dealing with fractions since direct differentiation isn't straightforward with a fraction. For the given function \( f(x) = \frac{2x^5 + x^2}{x} \), applying the Quotient Rule involved identifying:

• \( u(x) = 2x^5 + x^2 \)
• \( v(x) = x \)

Differentiating these, \( u'(x) = 10x^4 + 2x \) and \( v'(x) = 1 \), and substituting in the formula gives the derivative \( f'(x) = 8x^3 + 1 \). Differentiation is not just a mechanical tool, but it's crucial for understanding change in mathematics.

Function Simplification

Function simplification involves rewriting a complex expression in a simpler form. For the given exercise, we saw that the function \( f(x) = \frac{2x^5 + x^2}{x} \) could be simplified easily by dividing each term in the numerator by \( x \). This leads to a new, cleaner form:

• \( f(x) = 2x^4 + x \)

Simplifying functions when possible before differentiating can make the entire process much easier and faster. It eliminates the complexity of having to use advanced rules such as the Quotient Rule, where the denominator is a simple variable. In this problem, by simplifying first, the resulting derivative was found as \( f'(x) = 8x^3 + 1 \), consistent with the result from the Quotient Rule. Simplification can serve as a quick check for your work, helping confirm that your final answer is correct.

Graphing Calculator Verification

Using a graphing calculator is an excellent way to verify your calculations and visualise the behavior of functions and their derivatives. After calculating the derivative \( f'(x) = 8x^3 + 1 \) manually, inputting \( f(x) = \frac{2x^5 + x^2}{x} \) into a graphing calculator can confirm our results. With a graphing calculator, you can:

• Plot the original function \( f(x) \) and the derivative \( f'(x) \).
• Check the slope at particular points in the graph to see if it aligns with \( f'(x) = 8x^3 + 1 \).

These calculators typically offer a feature to numerically compute derivatives; hence, cross-referencing the slope with the calculated derivative illustrates your computation's accuracy. Graphing calculators are powerful learning tools, as they provide both a mathematical and visual confirmation of the differentiation process.