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.
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x + √xy = 6, (4, 1)
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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 3.3.2
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x² + x + 8
검증된 단계별 안내1
Step 1: Identify the function for which you need to find the derivatives. In this case, the function is y = x² + x + 8.
Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that if y = xⁿ, then the derivative dy/dx = n*x^(n-1).
Step 3: Apply the power rule to the first term x². The derivative of x² is 2*x^(2-1) = 2x.
Step 4: Apply the power rule to the second term x. The derivative of x is 1*x^(1-1) = 1.
Step 5: The constant term 8 has a derivative of 0, as the derivative of any constant is zero. Combine the derivatives of each term to find the first derivative: dy/dx = 2x + 1. Then, apply the same process to find the second derivative by differentiating dy/dx = 2x + 1 again.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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. In practical terms, the derivative provides the slope of the tangent line to the curve of the function at any given point.
추천 영상:
가이드 코스
05:44
Derivatives
Power Rule
The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if a function is in the form of f(x) = x^n, where n is a real number, then its derivative f'(x) is given by f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for functions involving powers of x.
추천 영상:
5:50Power Rules
Second Derivative
The second derivative of a function is the derivative of the first derivative. It provides information about the curvature of the function's graph and can indicate whether the function is concave up or concave down. The second derivative is useful for analyzing the acceleration of a function and identifying points of inflection where the function changes concavity.
추천 영상:
06:02The Second Derivative Test: Finding Local Extrema
관련 실천
교과서 질문
[Technology Exercise]
Graph the curves in Exercises 39–48.
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.
y = 4x²/⁵ − 2x
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교과서 질문
Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?
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교과서 질문
In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?
54. x³ + y² = sin²y
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교과서 질문
Derivatives in Differential Form
In Exercises 17–28, find dy.
xy² − 4x³/² − y = 0
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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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