교과서 질문
Assume that y = 5x and dx/dt = 2. Find dy/dt
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
3장, 문제 3.6.43
In Exercises 41–58, find dy/dt.
y = (1 + cos(2t))⁻⁴
검증된 단계별 안내1
Identify the function y = (1 + cos(2t))^(-4) and recognize that you need to find the derivative dy/dt.
Apply the chain rule for differentiation. The chain rule states that if you have a composite function y = f(g(t)), then dy/dt = f'(g(t)) * g'(t).
Differentiate the outer function f(u) = u^(-4) with respect to u, which gives f'(u) = -4u^(-5).
Substitute u = 1 + cos(2t) into the derivative of the outer function, resulting in -4(1 + cos(2t))^(-5).
Differentiate the inner function g(t) = 1 + cos(2t) with respect to t. The derivative of cos(2t) is -2sin(2t), so g'(t) = -2sin(2t). Multiply this result by the derivative of the outer function to get dy/dt = -4(1 + cos(2t))^(-5) * (-2sin(2t)).
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주요 개념
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Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(t)) is composed of two functions, then its derivative dy/dt is f'(g(t)) * g'(t). This rule is essential for finding the derivative of y = (1 + cos(2t))⁻⁴, as it involves differentiating the outer function and the inner function separately.
추천 영상:
05:02
Intro to the Chain Rule
Trigonometric Derivatives
Understanding the derivatives of trigonometric functions is crucial for solving this problem. The derivative of cos(t) is -sin(t), and when dealing with cos(2t), the chain rule must be applied, resulting in the derivative -2sin(2t). This knowledge is necessary to differentiate the inner function of the composite function y = (1 + cos(2t))⁻⁴.
추천 영상:
06:35Derivatives of Other Inverse Trigonometric Functions
Power Rule
The power rule is used to differentiate functions of the form y = x^n, where the derivative is n*x^(n-1). In this problem, the function y = (1 + cos(2t))⁻⁴ can be differentiated using the power rule, treating the entire expression inside the parentheses as a single variable. This rule helps in finding the derivative of the outer function in the composite expression.
추천 영상:
가이드 코스
5:50Power Rules
관련 실천
교과서 질문
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sin(5√x)
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교과서 질문
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.
2√y = x – y
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교과서 질문
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
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교과서 질문
Find the first and second derivatives of the functions in Exercises 33–38.
s = (t² + 5t − 1) / t²
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교과서 질문
In Exercises 43–50, find by implicit differentiation.
y² = x .
x + 1
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