Chapter 3: Problem 22
In Exercises \(15-22,\) find \(d y / d x\) . Support your answer graphically. $$y=\frac{(x+1)(x+2)}{(x-1)(x-2)}$$
Short Answer
Expert verified
The derivative of \(y = \frac{(x+1)(x+2)}{(x-1)(x-2)}\) is \(y' = \frac{-2x^2 + 4x + 2}{(x - 1)^2 (x - 2)^2}\).
Step by step solution
01
Apply the quotient rule
Differentiate the given function using quotient rule. The quotient rule states that the derivative of \(y = \frac{u}{v}\) is \(y' = \frac{u'v - uv'}{v^2}\). In this case, \(u = (x + 1)(x + 2)\) and \(v = (x - 1)(x - 2)\). First, find \(u'\) and \(v'\), the derivatives of \(u\) and \(v\) respectively.
02
Find the derivative of u and v
The derivative, \(u' = (x + 1)'(x + 2) + (x + 1)(x + 2)'\), results in \(u' = (1)(x + 2) + (x + 1)(1) = x + 2 + x + 1 = 2x + 3\). Similarly for \(v\), \(v' = (x - 1)'(x - 2) + (x - 1)(x - 2)'\), which gives \(v' = (1)(x - 2) + (x - 1)(1) = x - 2 + x - 1 = 2x - 3\).
03
Apply the quotient rule and simplify
Now apply the quotient rule: \(y' = \frac{u'v - uv'}{v^2} = \frac{(2x + 3)(x - 1)(x - 2) - (x + 1)(x + 2)(2x - 3)}{(x - 1)^2 (x - 2)^2}\). Simplify by expanding and combining like terms to get \(y' = \frac{-2x^2 + 4x + 2}{(x - 1)^2 (x - 2)^2}\).
04
Interpret the graph
Graphically, this will look like a curve for the original function \(y = \frac{(x + 1)(x + 2)}{(x - 1)(x - 2)}\) and a curve for the derived function \(y' = \frac{-2x^2 + 4x + 2}{(x - 1)^2 (x - 2)^2}\). Where the slope of the \(y\) curve is steepest, the \(y'\) function will have high positive or high negative values.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus
Calculus is a branch of mathematics focused on the study of change and motion. It's divided into two main areas: differential calculus and integral calculus. Differential calculus concerns itself with finding the rate at which quantities change, which is where the concept of the derivative comes into play. Meanwhile, integral calculus is about accumulation and area under curves. Together, these tools are widely used in science, engineering, economics, and many other fields to model and solve real-world problems.
One fundamental aspect of calculus is the ability to predict future events based on models of change. Therefore, mastering the concepts of differential and integral calculus equips students with the analytical skills required to tackle complex problems and understand how different quantities interact with each other over time.
One fundamental aspect of calculus is the ability to predict future events based on models of change. Therefore, mastering the concepts of differential and integral calculus equips students with the analytical skills required to tackle complex problems and understand how different quantities interact with each other over time.
Derivative Calculation
Derivative calculation is at the heart of differential calculus. It involves finding the derivative of a function, which represents the rate of change of the function's output with respect to changes in its input. In practical terms, derivatives can be seen as slopes of tangent lines to curves on a graph, and they help in understanding the behavior of functions by indicating points of increasing or decreasing values, as well as identifying maxima, minima, and inflection points.
The process of finding a derivative can be approached using various rules and techniques. The most commonly used ones include the power rule, product rule, quotient rule, and chain rule. In the context of an exercise like the one provided, the quotient rule is applied when differentiating a function that is a quotient of two other functions. Understanding when and how to apply these rules is essential for accurate and efficient derivative calculation.
The process of finding a derivative can be approached using various rules and techniques. The most commonly used ones include the power rule, product rule, quotient rule, and chain rule. In the context of an exercise like the one provided, the quotient rule is applied when differentiating a function that is a quotient of two other functions. Understanding when and how to apply these rules is essential for accurate and efficient derivative calculation.
Graphical Interpretation of Derivatives
The graphical interpretation of derivatives adds a visual dimension to understanding calculus. When looking at the graph of a function, its derivative can be graphically interpreted as the slope of the curve at any given point. In other words, it shows the steepness and direction of a curve's incline or decline at specific instances.
The derivative of a function at a point can be positive, negative, or zero, which has graphical implications. A positive derivative indicates that the function is increasing at that point, and graphically, the slope of the tangent line is upward. Conversely, a negative derivative suggests a decreasing function and a downward slope. A zero derivative implies that the function has a horizontal tangent line at that point, possibly indicating a local maximum or minimum.
Graphing the original function alongside its derivative provides insights into the behavior of the function, such as identifying where it is increasing or decreasing most rapidly. This visual understanding complements the analytical techniques used in derivative calculation.
The derivative of a function at a point can be positive, negative, or zero, which has graphical implications. A positive derivative indicates that the function is increasing at that point, and graphically, the slope of the tangent line is upward. Conversely, a negative derivative suggests a decreasing function and a downward slope. A zero derivative implies that the function has a horizontal tangent line at that point, possibly indicating a local maximum or minimum.
Graphing the original function alongside its derivative provides insights into the behavior of the function, such as identifying where it is increasing or decreasing most rapidly. This visual understanding complements the analytical techniques used in derivative calculation.
Chain Rule
The chain rule is a powerful derivative technique used in calculus to find the derivative of composite functions. When a function consists of a 'chain' of functions nested inside one another, the chain rule allows differentiation by taking the derivative of the outer function and multiplying it by the derivative of the inner function(s).
The rule essentially decomposes the overall rate of change into the product of rates corresponding to each link in the 'chain' of functions. It is particularly beneficial when dealing with complex functions that involve multiple compositions, such as trigonometric functions wrapped around polynomial expressions, or exponential functions with variable exponents.
Although the exercise at hand does not explicitly require the use of the chain rule, it's an important concept for students to understand, as they may encounter compositions of functions that necessitate the chain rule in more advanced problems.
The rule essentially decomposes the overall rate of change into the product of rates corresponding to each link in the 'chain' of functions. It is particularly beneficial when dealing with complex functions that involve multiple compositions, such as trigonometric functions wrapped around polynomial expressions, or exponential functions with variable exponents.
Although the exercise at hand does not explicitly require the use of the chain rule, it's an important concept for students to understand, as they may encounter compositions of functions that necessitate the chain rule in more advanced problems.
