Skip to main content
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 4.6.66
Chapter 4, Problem 4.6.66

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) = (x+4)/(4-x)

Verified step by step guidance
1
First, identify the function given: \( f(x) = \frac{x+4}{4-x} \). We need to find the derivative \( f'(x) \) to express the differential \( dy \).
Apply the quotient rule for differentiation, which states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = x+4 \) and \( v(x) = 4-x \).
Calculate the derivatives: \( u'(x) = 1 \) and \( v'(x) = -1 \).
Substitute these into the quotient rule formula: \( f'(x) = \frac{(1)(4-x) - (x+4)(-1)}{(4-x)^2} \). Simplify the expression to find \( f'(x) \).
Finally, express the differential relationship as \( dy = f'(x)dx \), where \( f'(x) \) is the simplified derivative from the previous step.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Differentials

Differentials represent the relationship between small changes in variables. In calculus, if y is a function of x, the differential dy is defined as dy = f'(x)dx, where f'(x) is the derivative of the function. This concept allows us to approximate how a small change in x (denoted as dx) affects the change in y (denoted as dy).
Recommended video:
05:53
Finding Differentials

Derivatives

The derivative of a function 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 the context of the given function, f'(x) will provide the slope of the tangent line at any point on the curve, which is essential for calculating dy.
Recommended video:
05:44
Derivatives

Function Composition

Function composition involves combining two functions to create a new function. In this case, understanding how the function f(x) = (x+4)/(4-x) behaves is crucial for finding its derivative. Analyzing the function's structure helps in applying the quotient rule for differentiation, which is necessary for determining f'(x) and subsequently expressing dy in terms of dx.
Recommended video:
3:48
Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question

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


lim_x→π/2⁻ (π/2 - x) sec x

1
views
Textbook Question

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


∫ (4√x - (4 /√x)) dx

Textbook Question

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.


y = 1/x and y = 4 - x²

Textbook Question

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.


g(s) = 1 / (s² + 1)

Textbook Question

Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).

Textbook Question

Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)