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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1:52 minutes

Problem 3.9.24

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = cos(x²)

Verified step by step guidance

Step 1: Identify the function y = cos(x²). This is a composition of functions, where the outer function is cos(u) and the inner function is u = x².

Step 2: Apply the chain rule for differentiation, which states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Step 3: Differentiate the outer function with respect to its argument u. The derivative of cos(u) with respect to u is -sin(u).

Step 4: Differentiate the inner function u = x² with respect to x. The derivative of x² with respect to x is 2x.

Step 5: Combine the results from steps 3 and 4 using the chain rule: dy/dx = -sin(x²) * 2x. This is the derivative of y with respect to x.

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Key Concepts

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

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative dy/dx is found by multiplying the derivative of the outer function f with respect to its inner function g, by the derivative of the inner function g with respect to x. This is essential for differentiating y = cos(x²), where x² is the inner function.

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions is crucial for solving problems involving these functions. The derivative of cos(x) with respect to x is -sin(x). This knowledge is applied when differentiating y = cos(x²), where the derivative of the outer function cos is needed to apply the chain rule effectively.

Differential Notation

Differential notation involves expressing the derivative of a function in terms of differentials, such as dy and dx. It provides a way to represent the rate of change of y with respect to x. In the context of the problem, finding dy involves using differential notation to express the derivative of y = cos(x²) in terms of dy and dx, which is useful for understanding changes in y as x varies.

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Related Practice

Textbook Question

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

g(t) = 1/(1 − t) + √(1 + t) − 3.1, (−1, 1)

Textbook Question

Finding Functions from Derivatives

Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Textbook Question

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.

c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)

Textbook Question

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

26. Factoring a quartic Find the approximate values of r_1 through r_4 in the factorization

8x^4-14x^3-9x^2+11x-1=8(x-r_1)(x-r_2)(x-r_3)(x-r_4)

Textbook Question

Dependence on Initial Point

8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations

c. x_0=2

Multiple Choice

If $f\left(x\right)=x^3-8x+6$ , find the differential $dy$ when x = 2 and $dx = 0.2$ .