교과서 질문
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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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
3장, 문제 3.5.15
Derivatives
In Exercises 1–18, find dy/dx.
y = (sec x + tan x)(sec x − tan x)
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
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.
추천 영상:
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.
추천 영상:
06:35Derivatives of Other Inverse Trigonometric Functions
관련 실천
교과서 질문
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.
__
x + √xy = 6, (4, 1)
288
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교과서 질문
In Exercises 19–22, find the slope of the curve at the point indicated.
y = x³ − 2x + 7, x = −2
245
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교과서 질문
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)
240
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교과서 질문
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?
186
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교과서 질문
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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