교과서 질문
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
g(x) = { x²/³, x ≥ 0
x¹/³, x < 0
219
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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 3.7.14
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x cos(2x + 3y) = y sin x
검증된 단계별 안내1
Start by differentiating both sides 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 left side: For x cos(2x + 3y), use the product rule. The derivative of x is 1, and the derivative of cos(2x + 3y) is -sin(2x + 3y) multiplied by the derivative of the inside function, which is 2 + 3(dy/dx).
Differentiate the right side: For y sin x, use the product rule again. The derivative of y is dy/dx, and the derivative of sin x is cos x.
Set the derivatives from both sides equal to each other. This will give you an equation involving dy/dx.
Solve the resulting equation for dy/dx to find the derivative of y with respect to x.
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.
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, often x, while treating other variables, like y, as implicit functions of x. This method is essential when dealing with equations where y cannot be easily 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. In implicit differentiation, the chain rule is often applied when differentiating terms involving y, as y is considered a function of x.
추천 영상:
05:02Intro to the Chain Rule
Trigonometric Derivatives
Trigonometric derivatives are formulas used to find the derivatives of trigonometric functions such as sine, cosine, and tangent. For example, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). In implicit differentiation problems involving trigonometric functions, these derivatives are crucial for correctly differentiating terms like x cos(2x + 3y) and y sin x.
추천 영상:
06:35Derivatives of Other Inverse Trigonometric Functions
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y = x⁵ / 120
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Find the derivatives of the functions in Exercises 1–42.
𝔂 = 1 x² csc 2
2 x
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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.
x²y² = 9, (–1,3)
232
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