Problem 28 - Brief Applied Calculus

Question

Find the derivative of each function in two ways: a. Using the Quotient rule. b. Simplifying the original function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ \frac{x^{9}}{x^{3}} $$

Step-by-Step Solution

Answer

The derivative is \(6x^5\) using both methods.

Step 1: Apply the Quotient Rule

The Quotient Rule states that if you have a function \( f(x) = \frac{u(x)}{v(x)} \), then the derivative \( f'(x) \) is given by\[f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}.\]For \( f(x) = \frac{x^9}{x^3} \), let \( u(x) = x^9 \) and \( v(x) = x^3 \). Then, \( u'(x) = 9x^8 \) and \( v'(x) = 3x^2 \).Substitute these into the Quotient Rule:\[f'(x) = \frac{9x^8 \cdot x^3 - x^9 \cdot 3x^2}{(x^3)^2}.\] Simplify the expression:\[= \frac{9x^{11} - 3x^{11}}{x^6} = \frac{6x^{11}}{x^6}.\]Finally, simplify further to\[f'(x) = 6x^{5}.\]

Step 2: Simplify Original Function

The original function \( f(x) = \frac{x^9}{x^3} \) can be simplified by dividing the exponents since both the numerator and the denominator are powers of \( x \).\[ f(x) = x^{9-3} = x^6.\]

Step 3: Apply the Power Rule

Now that we have \( f(x) = x^6 \), apply the Power Rule, which states \( \frac{d}{dx} x^n = nx^{n-1} \).Thus, the derivative is:\[f'(x) = 6x^{6-1} = 6x^5.\]

Step 4: Verify the Results Match

From both methods, using the Quotient Rule and simplifying with the Power Rule, we obtained the derivative:\[f'(x) = 6x^5.\] This confirms that both methods yield the same result.

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is essential for finding the derivative of a quotient of two functions. This rule tells us how to differentiate a function of the form \( f(x) = \frac{u(x)}{v(x)} \). It helps when you have a fraction where both the top and the bottom are functions of \( x \).

• To apply the Quotient Rule, you first find the derivatives of the numerator \( u(x) \) and the denominator \( v(x) \).
• According to the rule, the derivative is given by: \[f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}.\]
• This formula helps you find the change rate for the entire fraction.

Using the rule requires careful algebra to simplify expressions afterward. For the given problem, applying the Quotient Rule to \( \frac{x^9}{x^3} \), we worked out that both the numerator and denominator functions needed separate derivatives before plugging them into the formula.

Power Rule

The Power Rule is one of the simplest and most useful tools for finding derivatives, making it crucial for any calculus student. It applies to functions that are powers of \( x \), written as \( x^n \), where \( n \) is any real number.

• The rule itself is straightforward: \[\frac{d}{dx} x^n = nx^{n-1}.\]
• Simply multiply by the exponent and subtract one from the original exponent.

In the given exercise, after simplifying the function \( \frac{x^9}{x^3} \) to \( x^6 \), the Power Rule allows us to quickly find the derivative as \( 6x^5 \).This shows how efficient the Power Rule is compared to more complex rules like the Quotient Rule when applicable.

Derivative

The derivative represents the rate of change of a function as its input changes. It is foundational in calculus as it provides insight into how functions behave and change over time or space.

• It is symbolized by \( f'(x) \) or \( \frac{df(x)}{dx} \).
• Derivatives can be calculated using various rules, including both the Quotient Rule and Power Rule.

In the problem presented, finding the derivative was about determining how the function \( \frac{x^9}{x^3} \) changes. Through both the Quotient and Power Rules, we found that the derivative is \( 6x^5 \).This consistency reinforces that different methods can yield the same outcome, proving their reliability and utility in various circumstances.