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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.6.69
Chapter 4, Problem 4.6.69

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = tan x

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First, understand that the differential dy represents the change in the function's output, while dx represents a small change in the input x.
To find the relationship between dy and dx, we need to compute the derivative of the function f(x) = tan(x).
Recall that the derivative of tan(x) with respect to x is sec^2(x). This is a standard derivative result.
Express the differential relationship using the formula dy = f'(x)dx. Substitute f'(x) with sec^2(x) to get dy = sec^2(x)dx.
This equation, dy = sec^2(x)dx, shows how a small change in x (dx) affects the change in y (dy) for the function f(x) = tan(x).

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Key Concepts

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

Differentials

Differentials represent the infinitesimal changes in variables. In calculus, if y = f(x), the differential dy is defined as dy = f'(x)dx, where f'(x) is the derivative of f with respect to x. This relationship allows us to approximate how a small change in x (denoted as dx) affects the change in y (denoted as dy).
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Finding Differentials

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 as the interval approaches zero. For the function f(x) = tan x, the derivative f'(x) can be calculated using differentiation rules, which will be essential for expressing dy in terms of dx.
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Derivatives

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. Understanding the chain rule is crucial when dealing with functions like f(x) = tan x, especially when applying it to find the relationship between dx and dy.
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Related Practice
Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (sec² x - 1) dx

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)

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Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = e⁻ˣ - ((x + 4)/5)

Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = cos 2x - x² + 2x

Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = cos² x on [-π,π]

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Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (eˣ - 1) / (2x + 5)