Textbook Question
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
xy + 2x - 5y = 2, (3, 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 3.9.27
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = 3 csc(1 − 2√x)
Verified step by step guidance1
Step 1: Identify the function y = 3 csc(1 - 2√x). The goal is to find dy, which involves differentiating y with respect to x.
Step 2: Recognize that the function involves a composite function: y = 3 csc(u), where u = 1 - 2√x. Use the chain rule for differentiation, which states that dy/dx = dy/du * du/dx.
Step 3: Differentiate the outer function with respect to u. The derivative of csc(u) with respect to u is -csc(u)cot(u). Therefore, dy/du = 3 * (-csc(u)cot(u)).
Step 4: Differentiate the inner function u = 1 - 2√x with respect to x. The derivative of √x is 1/(2√x), so du/dx = -2 * (1/(2√x)) = -1/√x.
Step 5: Combine the derivatives using the chain rule: dy/dx = dy/du * du/dx = 3 * (-csc(u)cot(u)) * (-1/√x). Substitute u = 1 - 2√x back into the expression to complete the differentiation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Trigonometric Functions
Understanding the derivatives of trigonometric functions is crucial. The derivative of csc(x) is -csc(x)cot(x). This knowledge helps in differentiating expressions involving trigonometric functions, such as y = 3 csc(1 - 2√x), by applying the chain rule to account for the inner function.
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Introduction to Trigonometric Functions
Chain Rule
The chain rule is essential for differentiating composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x))g'(x). In the given problem, the chain rule helps differentiate the inner function 1 - 2√x, which is part of the composite trigonometric function.
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05:02Intro to the Chain Rule
Differentiation of Radical Functions
Differentiating functions involving radicals, such as √x, requires understanding their derivatives. The derivative of √x is (1/2)x^(-1/2). This concept is necessary to find the derivative of the inner function 1 - 2√x, which is part of the overall differentiation process in the given problem.
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05:53Finding Differentials
Related Practice
Textbook Question
Find the tangent line to the Witch of Agnesi (graphed here) at the point (2,1).
Textbook Question
In Exercises 43–50, find by implicit differentiation.
__
√xy = 1
Textbook Question
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.
y² = x² + 2x
Textbook Question
In Exercises 65 and 66, find the derivative using the definition.
ƒ(t) = 1 .
2t + 1
Textbook Question
In Exercises 29 and 30, find the slope of the curve at the given points.
y² + x² = y⁴ – 2x at (–2,1) and (–2,–1)