Textbook Question
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
f(x) = { 2x − 1, x ≥ 0
x² + 2x + 7, x < 0
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 3.5.15
Derivatives
In Exercises 1–18, find dy/dx.
y = (sec x + tan x)(sec x − tan x)
Verified step by step guidance1
Recognize that the expression y = (sec x + tan x)(sec x − tan x) is a product of two functions. To find dy/dx, we will use the product rule for differentiation, which states that if y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v'(x).
Identify the two functions: u(x) = sec x + tan x and v(x) = sec x − tan x.
Differentiate u(x) with respect to x. The derivative of sec x is sec x tan x, and the derivative of tan x is sec^2 x. Therefore, u'(x) = sec x tan x + sec^2 x.
Differentiate v(x) with respect to x. Similarly, v'(x) = sec x tan x - sec^2 x.
Apply the product rule: dy/dx = (sec x tan x + sec^2 x)(sec x − tan x) + (sec x + tan x)(sec x tan x - sec^2 x). Simplify the expression by expanding and combining like terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product Rule
The Product Rule is a fundamental differentiation technique used when finding the derivative of a product of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by d(uv)/dx = u'v + uv'. This rule is essential for the given problem, as the expression involves the product of two binomials.
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The Product Rule
Chain Rule
The Chain Rule is another critical differentiation principle that allows us to differentiate composite functions. If a function y is defined as a composition of two functions, say y = f(g(x)), the Chain Rule states that dy/dx = f'(g(x)) * g'(x). This concept may be relevant if the functions within the product involve more complex expressions that require differentiation of inner functions.
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05:02Intro to the Chain Rule
Trigonometric Derivatives
Understanding the derivatives of trigonometric functions is crucial for solving problems involving secant and tangent functions. The derivatives are given by d(sec x)/dx = sec x tan x and d(tan x)/dx = sec^2 x. Recognizing these derivatives will facilitate the differentiation of the given expression, allowing for a more straightforward application of the Product Rule.
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06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x⁻¹/² sec (2x)²
Textbook Question
Second Derivatives
Find y'' in Exercises 59–64.
y = x(2x + 1)⁴
Textbook Question
Find an equation of the straight line having slope 1/4 that is tangent to the curve y = √x.
Textbook Question
Find the derivatives of the functions in Exercises 17–28.
y = ((x + 1)(x + 2)) / ((x − 1)(x − 2))
Textbook Question
If r + s² + v³ = 12, dr/dt = 4, and ds/dt = –3, find dv/dt when r = 3 and s = 1.