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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2:14 minutes

Problem 4.6.69

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) = tan x

Verified step by step guidance

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).

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.

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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