Textbook Question
5–8. Calculate dy/dx using implicit differentiation.
sin y+2 = x
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.5.70
Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/4 cot x−1 / x−π/4
Verified step by step guidance1
Recognize that the given limit can be interpreted as the derivative of a function at a point. Specifically, it resembles the definition of the derivative of the function f(x) = cot(x) at the point x = π/4.
Recall the definition of the derivative: f'(a) = lim x→a (f(x) - f(a)) / (x - a). In this case, f(x) = cot(x) and a = π/4, so f(a) = cot(π/4) = 1.
Substitute f(x) = cot(x) and f(a) = 1 into the derivative definition: f'(π/4) = lim x→π/4 (cot(x) - 1) / (x - π/4). This matches the given limit expression.
To find the derivative f'(x) of f(x) = cot(x), use the derivative formula: f'(x) = -csc^2(x). Evaluate this derivative at x = π/4.
Calculate f'(π/4) using the derivative formula: f'(π/4) = -csc^2(π/4). Since csc(π/4) = √2, f'(π/4) = -(√2)^2 = -2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches π/4 helps determine the behavior of the function cot(x) - 1 near that point.
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Derivatives
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this problem, the limit can be interpreted as the derivative of the function cot(x) at x = π/4.
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Cotangent Function
The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is important to understand its behavior, especially around specific angles like π/4, where cot(π/4) equals 1. This knowledge is crucial for evaluating the limit in the given problem.
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Related Practice
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Textbook Question
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = x^2/3, for x>0