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Magazine Chapter 4: Problem 22
In Exercises 13 through \(24,\) use the quotient rule to find the derivative. $$ f(x)=\frac{2 x-3}{4 x-1} $$
Short Answer
Expert verified
\( f'(x) = \frac{10}{(4x - 1)^2} \).
Step by step solution
01
Identify the Functions
In the quotient \( f(x) = \frac{2x - 3}{4x - 1} \), identify the numerator and denominator functions. Here, \( u(x) = 2x - 3 \) and \( v(x) = 4x - 1 \).
02
Derivative of the Numerator
Find the derivative of the numerator function: \( u'(x) = \frac{d}{dx}(2x - 3) = 2 \).
03
Derivative of the Denominator
Find the derivative of the denominator function: \( v'(x) = \frac{d}{dx}(4x - 1) = 4 \).
04
Apply the Quotient Rule
The quotient rule states \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \). Substitute \( u(x) \), \( v(x) \), \( u'(x) \), and \( v'(x) \) into the formula. \( f'(x) = \frac{2(4x - 1) - (2x - 3)(4)}{(4x - 1)^2} \).
05
Simplify the Expression
Simplify the expression: \[ f'(x) = \frac{8x - 2 - (8x - 12)}{(4x - 1)^2} = \frac{8x - 2 - 8x + 12}{(4x - 1)^2} = \frac{10}{(4x - 1)^2} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative
The concept of a derivative is a foundational idea in calculus that deals with the rate of change of a function. If you're analyzing how something changes, like the speed of a car or the slope of a hill, you're essentially dealing with derivatives. In the case of functions like \( f(x) = \frac{2x - 3}{4x - 1} \), the derivative helps us understand how the output of the function changes as the input \( x \) changes.
Calculating a derivative requires careful attention to mathematical principles. Here, understanding how to apply the quotient rule accurately is key to determining how the function's behavior changes locally.
- To find the derivative efficiently, we employ the derivative rules of calculus, like the power rule, product rule, and quotient rule.
- Derivatives are essential in various fields, from physics to economics, because they can describe dynamics within those disciplines.
- In this exercise, the quotient rule is utilized to differentiate a rational function, showcasing the calculated approach to finding derivatives of complex expressions.
Calculating a derivative requires careful attention to mathematical principles. Here, understanding how to apply the quotient rule accurately is key to determining how the function's behavior changes locally.
Calculus
Calculus is the branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It's a comprehensive toolkit for analyzing change and motion. Calculus can be broadly divided into differential calculus, which deals with rates of change, and integral calculus, which involves accumulation of quantities.
By applying calculus, we can understand and predict the behavior of various phenomena:
The understanding developed through calculus is crucial for advanced studies in fields that rely heavily on statistical data analysis and interpretation of dynamic systems.
By applying calculus, we can understand and predict the behavior of various phenomena:
- It support analyses of physical systems, economic models, and engineering structures.
- In this problem, we specifically look at differential calculus to understand how a rational function changes.
- The use of calculus allows us to find not just how quickly things change, but also to find the extremes, like maximum or minimum values, through its applications.
The understanding developed through calculus is crucial for advanced studies in fields that rely heavily on statistical data analysis and interpretation of dynamic systems.
Rational Functions
A rational function is a fraction where both the numerator and the denominator are polynomials. In mathematics, these functions demonstrate significant properties and behaviors due to their form. For the given function \( f(x) = \frac{2x - 3}{4x - 1} \), the key task is handling its rates of change and characteristics:
Rational functions have unique properties, such as asymptotic behavior and points of discontinuity. Understanding how to manipulate and analyze these functions is necessary for deeper mathematical investigations, as well as practical applications, such as modeling real-world scenarios.
- The numerator \( u(x) = 2x - 3 \) and denominator \( v(x) = 4x - 1 \) are polynomials.
- Analyzing such functions involves investigating constraints, like when the denominator equals zero, producing vertical asymptotes.
- The quotient rule is particularly useful for taking derivatives of rational functions, making the analysis of these behaviors manageable.
Rational functions have unique properties, such as asymptotic behavior and points of discontinuity. Understanding how to manipulate and analyze these functions is necessary for deeper mathematical investigations, as well as practical applications, such as modeling real-world scenarios.
