Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.5.42

Chapter 3, Problem 3.5.42

23–51. Calculating derivatives Find the derivative of the following functions.
y = tan x + cot x

Verified step by step guidance

Step 1: Identify the functions involved in the expression. The given function is \( y = \tan x + \cot x \). Here, \( \tan x \) and \( \cot x \) are trigonometric functions.

Step 2: Recall the derivatives of the basic trigonometric functions. The derivative of \( \tan x \) is \( \sec^2 x \), and the derivative of \( \cot x \) is \( -\csc^2 x \).

Step 3: Apply the sum rule for derivatives. The sum rule states that the derivative of a sum of functions is the sum of their derivatives. Therefore, \( \frac{d}{dx}(\tan x + \cot x) = \frac{d}{dx}(\tan x) + \frac{d}{dx}(\cot x) \).

Step 4: Substitute the derivatives of the individual functions into the expression. This gives \( \sec^2 x - \csc^2 x \).

Step 5: Simplify the expression if possible. In this case, the expression \( \sec^2 x - \csc^2 x \) is already in its simplest form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate at which a function changes at any given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, are functions of an angle that relate the angles of a triangle to the lengths of its sides. In calculus, understanding the derivatives of these functions is crucial, as they often appear in various mathematical models and applications.

Derivative Rules

Derivative rules, such as the sum rule, product rule, and quotient rule, provide systematic methods for finding the derivatives of complex functions. The sum rule states that the derivative of a sum of functions is the sum of their derivatives, which is particularly useful when differentiating functions like y = tan x + cot x.

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Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

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Textbook Question

The bottom of a large theater screen is 3 ft above your eye level and the top of the screen is 10 ft above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of 3 ft/s while looking at the screen. What is the rate of change of the viewing angle θ when you are 30 ft from the wall on which the screen hangs, assuming the floor is horizontal (see figure)? <IMAGE>

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Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

g(t) = 6√t

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Textbook Question

15–48. Derivatives Find the derivative of the following functions.

s(t) = cos 2^t

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