Textbook Question
Parallel tangent lines Find the two points where the curve x² + xy + y² = 7 crosses the x-axis, and show that the tangent lines to the curve at these points are parallel. What is the common slope of these tangent lines?
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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.
Recommended video:
05:18
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.
Recommended video:
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.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
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x + √xy = 6, (4, 1)
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Textbook Question
In Exercises 19–22, find the slope of the curve at the point indicated.
y = x³ − 2x + 7, x = −2
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Textbook Question
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
y = 2 sin(πx – y), (1,0)
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Textbook Question
The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s
b. volume?
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Textbook Question
Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².
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