교과서 질문
Find the first and second derivatives of the functions in Exercises 33–38.
w = ((1 + 3z) / 3z) (3 − z)
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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 3.7.20
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
x²/³ + y²/³ = 1
검증된 단계별 안내1
Start by differentiating both sides of the equation x^(2/3) + y^(2/3) = 1 with respect to x. Use implicit differentiation for the y term.
For the term x^(2/3), apply the power rule: differentiate to get (2/3)x^(-1/3).
For the term y^(2/3), apply the chain rule: differentiate to get (2/3)y^(-1/3) * (dy/dx).
Set the derivative of the left side equal to the derivative of the right side (which is 0), and solve for dy/dx.
To find the second derivative d²y/dx², differentiate the expression for dy/dx with respect to x, applying implicit differentiation again, and express the result in terms of x and y.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to a variable, typically x, while treating other variables as implicit functions of x. This method is essential for finding dy/dx when y is not isolated.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is crucial when applying implicit differentiation, as it allows for the differentiation of terms involving y with respect to x.
추천 영상:
05:02Intro to the Chain Rule
Second Derivative
The second derivative, denoted as d²y/dx², represents the derivative of the first derivative, providing information about the curvature or concavity of a function. In the context of implicit differentiation, finding the second derivative involves differentiating the expression for dy/dx again with respect to x, often requiring the use of the chain rule and implicit differentiation techniques.
추천 영상:
06:02The Second Derivative Test: Finding Local Extrema
관련 실천
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교과서 질문
Second Derivatives
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