Textbook Question
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
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
Chapter 3, Problem 3.6.52
In Exercises 41–58, find dy/dt.
y = (1/6)(1 + cos²(7t))³
Verified step by step guidance1
Identify the function y = \( \frac{1}{6} (1 + \cos^2(7t))^3 \) and recognize that you need to find \( \frac{dy}{dt} \). This involves using the chain rule for differentiation.
Apply the chain rule: If \( y = u^3 \) where \( u = 1 + \cos^2(7t) \), then \( \frac{dy}{dt} = 3u^2 \cdot \frac{du}{dt} \).
Differentiate \( u = 1 + \cos^2(7t) \) with respect to t. Use the chain rule again: \( \frac{du}{dt} = 2\cos(7t) \cdot (-\sin(7t)) \cdot 7 \).
Simplify \( \frac{du}{dt} \) to \( -14\cos(7t)\sin(7t) \). This can be further simplified using the double angle identity: \( \sin(2x) = 2\sin(x)\cos(x) \), so \( \frac{du}{dt} = -7\sin(14t) \).
Substitute \( u \) and \( \frac{du}{dt} \) back into the expression for \( \frac{dy}{dt} \): \( \frac{dy}{dt} = \frac{1}{6} \cdot 3(1 + \cos^2(7t))^2 \cdot (-7\sin(14t)) \). Simplify this expression to find the derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chain Rule
The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate the nested functions within y = (1/6)(1 + cos²(7t))³, particularly the inner function cos²(7t) and its outer cube.
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Intro to the Chain Rule
Derivative of Trigonometric Functions
Understanding how to differentiate trigonometric functions is crucial for solving this problem. The derivative of cos(x) is -sin(x), and when dealing with cos²(x), the power rule and chain rule are applied. Specifically, for cos²(7t), you need to differentiate using the chain rule, considering the inner function 7t and the outer square.
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Power Rule
The power rule is used to differentiate functions of the form x^n, where the derivative is n*x^(n-1). In this problem, the power rule is applied to the function (1 + cos²(7t))³, where the exponent 3 is brought down, and the function inside is raised to the power of 2, followed by differentiating the inner function using the chain rule.
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Related Practice
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