교과서 질문
Using the Alternative Formula for Derivatives
Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.
g(x) = 1 + √x
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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 3.9.21
Derivatives in Differential Form
In Exercises 17–28, find dy.
2y³/² + xy − x = 0
검증된 단계별 안내1
First, identify the given equation: 2y^(3/2) + xy - x = 0. We need to find dy, which involves differentiating the equation with respect to x.
Apply implicit differentiation to each term of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use the chain rule.
Differentiate the first term, 2y^(3/2), with respect to x. Using the chain rule, this becomes (3/2) * 2 * y^(1/2) * dy/dx.
Differentiate the second term, xy, with respect to x. This requires the product rule: differentiate x to get 1, and differentiate y to get dy/dx, resulting in y + x * dy/dx.
Differentiate the third term, -x, with respect to x, which simply becomes -1. Combine all differentiated terms and solve for dy/dx to find dy.
비슷한 문제에 대한 검증된 영상 답변:
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. In equations where y is not isolated, we differentiate both sides with respect to x, treating y as a function of x, and apply the chain rule to terms involving y.
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 differentiating terms like y^(3/2) with respect to x, where y is a function of x.
추천 영상:
05:02Intro to the Chain Rule
Solving for dy
After applying implicit differentiation, the goal is to solve for dy/dx, which represents the derivative of y with respect to x. This involves isolating dy/dx on one side of the equation, often requiring algebraic manipulation to combine like terms and factor out dy/dx, ultimately expressing it in terms of x and y.
추천 영상:
5:02Solving Logarithmic Equations
관련 실천
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The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).
203
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교과서 질문
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
g(x) = 8 / x², (2, 2)
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Find the derivatives of the functions in Exercises 19–40.
f(x) = √(7 + x sec x)
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Find the derivatives of the functions in Exercises 1–42.
𝔂 = x² sin² (2x²)
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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/cos²u, u = g(x) = πx, x = 1/4
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