Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x⁻¹/² sec (2x)²
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.3.28
Find the derivatives of the functions in Exercises 17–28.
y = ((x + 1)(x + 2)) / ((x − 1)(x − 2))
Verified step by step guidance1
Step 1: Identify the function type. The given function is a rational function, which is a ratio of two polynomials: \( y = \frac{(x + 1)(x + 2)}{(x - 1)(x - 2)} \).
Step 2: Apply the quotient rule for derivatives. The quotient rule states that if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \), where \( u = (x + 1)(x + 2) \) and \( v = (x - 1)(x - 2) \).
Step 3: Differentiate the numerator \( u = (x + 1)(x + 2) \). Use the product rule: \( u' = (x + 1)'(x + 2) + (x + 1)(x + 2)' \). Calculate \( (x + 1)' \) and \( (x + 2)' \).
Step 4: Differentiate the denominator \( v = (x - 1)(x - 2) \). Similarly, use the product rule: \( v' = (x - 1)'(x - 2) + (x - 1)(x - 2)' \). Calculate \( (x - 1)' \) and \( (x - 2)' \).
Step 5: Substitute \( u' \), \( v' \), \( u \), and \( v \) into the quotient rule formula \( y' = \frac{u'v - uv'}{v^2} \) to find the derivative of the function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. Derivatives are fundamental in calculus for understanding rates of change and are used in various applications, including optimization and motion analysis.
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Derivatives
Quotient Rule
The quotient rule is a formula used to find the derivative of a function that is the ratio of two other functions. If you have a function y = f(x)/g(x), the derivative is given by (g(x)f'(x) - f(x)g'(x)) / (g(x))^2. This rule is essential when differentiating functions that are expressed as fractions, as it allows for the correct application of the product and chain rules.
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Product Rule
The product rule is a technique for finding the derivative of a product of two functions. If y = f(x)g(x), the derivative is given by f'(x)g(x) + f(x)g'(x). This rule is crucial when dealing with functions that are multiplied together, ensuring that both functions' rates of change are accounted for in the overall derivative.
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Related Practice
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