Textbook Question
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x cos(2x + 3y) = y sin x
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 3.6.77
Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.
Verified step by step guidance1
Identify the given function: s = cos(θ). We need to find ds/dt, the rate of change of s with respect to time.
Use the chain rule for differentiation, which states that ds/dt = (ds/dθ) * (dθ/dt).
Differentiate s = cos(θ) with respect to θ to find ds/dθ. The derivative of cos(θ) is -sin(θ), so ds/dθ = -sin(θ).
Substitute the given value of dθ/dt = 5 into the chain rule expression: ds/dt = -sin(θ) * 5.
Evaluate -sin(θ) at θ = 3π/2. Since sin(3π/2) = -1, substitute this value into the expression to find ds/dt.
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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 concept in calculus used to differentiate composite functions. It states that if a variable y depends on u, which in turn depends on x, then the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x. In this problem, it helps find ds/dt by relating ds/dθ and dθ/dt.
Recommended video:
05:02
Intro to the Chain Rule
Trigonometric Derivatives
Understanding the derivatives of trigonometric functions is crucial for solving calculus problems involving these functions. The derivative of cos(θ) with respect to θ is -sin(θ). This knowledge is essential for finding ds/dθ when s = cos(θ), which is a step in applying the chain rule to find ds/dt.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Evaluating Trigonometric Functions
Evaluating trigonometric functions at specific angles is necessary for solving problems involving these functions. At θ = 3π/2, the value of sin(θ) is -1. This evaluation is crucial for determining the value of ds/dθ at the given angle, which is then used to find ds/dt using the chain rule.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
______
𝓻 = √2θ sinθ
Textbook Question
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)
Textbook Question
In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
y = (4 − 3x)⁹
Textbook Question
Find the derivatives of all orders of the functions in Exercises 29–32.
y = x⁵ / 120