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

Previous problemNext problem

3:01 minutes

Problem 3.9.27

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = 3 csc(1 − 2√x)

Verified step by step guidance

Step 1: Identify the function y = 3 csc(1 - 2√x). The goal is to find dy, which involves differentiating y with respect to x.

Step 2: Recognize that the function involves a composite function: y = 3 csc(u), where u = 1 - 2√x. Use the chain rule for differentiation, which states that dy/dx = dy/du * du/dx.

Step 3: Differentiate the outer function with respect to u. The derivative of csc(u) with respect to u is -csc(u)cot(u). Therefore, dy/du = 3 * (-csc(u)cot(u)).

Step 4: Differentiate the inner function u = 1 - 2√x with respect to x. The derivative of √x is 1/(2√x), so du/dx = -2 * (1/(2√x)) = -1/√x.

Step 5: Combine the derivatives using the chain rule: dy/dx = dy/du * du/dx = 3 * (-csc(u)cot(u)) * (-1/√x). Substitute u = 1 - 2√x back into the expression to complete the differentiation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions is crucial. The derivative of csc(x) is -csc(x)cot(x). This knowledge helps in differentiating expressions involving trigonometric functions, such as y = 3 csc(1 - 2√x), by applying the chain rule to account for the inner function.

Chain Rule

The chain rule is essential for differentiating composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x))g'(x). In the given problem, the chain rule helps differentiate the inner function 1 - 2√x, which is part of the composite trigonometric function.

Differentiation of Radical Functions

Differentiating functions involving radicals, such as √x, requires understanding their derivatives. The derivative of √x is (1/2)x^(-1/2). This concept is necessary to find the derivative of the inner function 1 - 2√x, which is part of the overall differentiation process in the given problem.

Watch next

Master Finding Differentials with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = x√(1 − x²)

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

2y³/² + xy − x = 0

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sec(x² − 1)

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

Textbook Question

Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

<IMAGE>

f(x) = x² + 2x, x₀ = 1, dx = 0.1

Textbook Question

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s

b. area?

Textbook Question

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s

a. circumference?

Textbook Question

Differential Estimates of Change

In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.

The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

Derivatives in Differential FormIn Exercises 17–28, find dy.y = 3 csc(1 − 2√x)

Key Concepts

Derivative of Trigonometric Functions

Chain Rule

Differentiation of Radical Functions

Watch next

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = 3 csc(1 − 2√x)