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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1:30 minutes

Problem 4.6.67

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) = 3x³ - 4x

Verified step by step guidance

First, identify the function given: \( f(x) = 3x^3 - 4x \). This is a polynomial function.

To find the differential \( dy \), we need to compute the derivative of \( f(x) \) with respect to \( x \). The derivative, \( f'(x) \), represents the rate of change of \( y \) with respect to \( x \).

Apply the power rule to differentiate \( f(x) \). The power rule states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).

Differentiate each term separately: For \( 3x^3 \), the derivative is \( 9x^2 \). For \( -4x \), the derivative is \( -4 \). Therefore, \( f'(x) = 9x^2 - 4 \).

Express the relationship between the small change in \( x \) and the corresponding change in \( y \) using differentials: \( dy = f'(x)dx = (9x^2 - 4)dx \). This equation shows how a small change in \( x \) results in a change in \( y \).

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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 infinitesimally small changes in variables. In calculus, the differential of a function, denoted as dy, indicates how much the function's output changes in response to a small change in its input, dx. This concept is foundational for understanding how functions behave locally and is crucial for applications in optimization and approximation.

Derivative

The derivative of a function, denoted as f'(x), 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. The derivative provides essential information about the function's slope and is used to find tangents, optimize functions, and analyze motion.

Chain Rule

The chain rule is a fundamental theorem in calculus that allows us 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 f with the derivative of the inner function g. This rule is essential for handling more complex functions and understanding how changes in one variable affect another.

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