교과서 질문
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = x√(1 − x²)
165
views
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
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 3.7.25
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.
3 + sin y = y – x³
검증된 단계별 안내1
Start by differentiating both sides of the equation 3 + sin(y) = y - x³ with respect to x. Remember that y is a function of x, so use implicit differentiation.
Differentiate the left side: The derivative of 3 is 0, and the derivative of sin(y) with respect to x is cos(y) * (dy/dx) using the chain rule.
Differentiate the right side: The derivative of y with respect to x is dy/dx, and the derivative of -x³ is -3x².
Set the derivatives equal: cos(y) * (dy/dx) = dy/dx - 3x². Solve this equation for dy/dx.
To find the second derivative d²y/dx², differentiate the expression for dy/dx with respect to x again. Use implicit differentiation and the chain rule where necessary, and express the result in terms of x and y.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
6m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where y is not explicitly solved for in terms of x. Instead of isolating y, we differentiate both sides of the equation with respect to x, applying the chain rule when differentiating y. This allows us to find dy/dx in terms of both x and y, which is essential for solving equations that define y implicitly.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative
First Derivative (dy/dx)
The first derivative, denoted as dy/dx, represents the rate of change of y with respect to x. It provides information about the slope of the tangent line to the curve defined by the equation at any point. In the context of implicit differentiation, finding dy/dx allows us to understand how y changes as x varies, which is crucial for further analysis, such as finding the second derivative.
추천 영상:
07:09The First Derivative Test: Finding Local Extrema
Second Derivative (d²y/dx²)
The second derivative, denoted as d²y/dx², measures the rate of change of the first derivative, indicating how the slope of the tangent line is changing. It provides insights into the concavity of the function and can reveal points of inflection. To find d²y/dx² using implicit differentiation, we differentiate dy/dx again, applying the product and chain rules, and express the result in terms of x and y.
추천 영상:
06:02The Second Derivative Test: Finding Local Extrema
관련 실천
교과서 질문
The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.
202
views
교과서 질문
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change
138
views
교과서 질문
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1
291
views
교과서 질문
In Exercises 41–58, find dy/dt.
y = sin²(πt − 2)
272
views
교과서 질문
Derivatives
In Exercises 23–26, find dr/dθ.
r = (1 + sec θ) sin θ
242
views