Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Differentials

Calculus

4. Applications of Derivatives

Differentials

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2:43 minutes

Problem 4.6.66

Textbook Question

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

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.

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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 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).

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.

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.

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