Q. 56 Use the definition of the deriva... [FREE SOLUTION]

91影视

Chapter 2: Q. 56 (page 184)

Use the definition of the derivative to find the derivatives described in Exercises 55-58.
Find ${\frac{d}{d x} (2 x^{3})|}_{3}, {\frac{d^{2}}{d x^{2}} (2 x^{3})|}_{1}, {\frac{d^{3}}{d x^{3}} (2 x^{3})|}_{- 2}$

Short Answer

Expert verified

The values are $54, 12, 12$ respectively

Step by step solution

Step 1. Given information

Given function $y = 2 x^{3}$

Use the definition of derivative and calculate

Calculating, we get

$\lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h} = \lim_{h 鈫� 0} \frac{2 (x + h)^{3} - 2 x^{3}}{h} \lim_{h 鈫� 0} \frac{f (x + h) - f (x)}{h} = = \lim_{h 鈫� 0} \frac{2 (x^{3} + h^{3} + 3 x^{2} h + 3 x h^{2}) - 2 x^{3}}{h} = \lim_{h 鈫� 0} \frac{2 x^{3} + 2 h^{3} + 6 x^{2} h + 6 x h^{2} - 2 x^{3}}{h} = \lim_{h 鈫� 0} \frac{2 h^{3} + 6 x^{2} h + 6 x h^{2}}{h} = 6 x^{2} 6 {(3)}^{2} = 6 脳 9 = 54$

Calculate other derivatives

Calculating, we get

$\lim_{h 鈫� 0} \frac{g (x + h) - g (x)}{h} = \lim_{h 鈫� 0} \frac{6 (x + h)^{2} - 6 x^{2}}{h} \lim_{h 鈫� 0} \frac{g (x + h) - g (x)}{h} = = \lim_{h 鈫� 0} \frac{6 (x^{2} + h^{2} + 2 x h) - 6 x^{2}}{h} = \lim_{h 鈫� 0} \frac{6 h^{2} + 12 x h}{h} = 12 x 12 (1) = 12$

Calculating other derivatives

Calculating, we get

$\lim_{h 鈫� 0} \frac{m (x + h) - m (x)}{h} = \lim_{h 鈫� 0} \frac{12 (x + h) - 12 x}{h} \lim_{h 鈫� 0} \frac{m (x + h) - m (x)}{h} = \lim_{h 鈫� 0} \frac{12 x + 12 h - 12 x}{h} = \lim_{h 鈫� 0} \frac{12 h}{h} = 12$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use the definition of the derivative to find the derivatives described in Exercises 55-58.
Find ${\frac{d}{d x} (2 x^{3})|}_{3}, {\frac{d^{2}}{d x^{2}} (2 x^{3})|}_{1}, {\frac{d^{3}}{d x^{3}} (2 x^{3})|}_{- 2}$

Short Answer

Step by step solution

Step 1. Given information

Use the definition of derivative and calculate

Calculate other derivatives

Calculating other derivatives

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Mechanics Maths

Calculus

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

91影视

Use the definition of the derivative to find the derivatives described in Exercises 55-58.Findddx2x33,d2dx22x31,d3dx32x3-2

Short Answer

Step by step solution

Step 1. Given information

Use the definition of derivative and calculate

Calculate other derivatives

Calculating other derivatives

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Mechanics Maths

Calculus

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use the definition of the derivative to find the derivatives described in Exercises 55-58.
Find ${\frac{d}{d x} (2 x^{3})|}_{3}, {\frac{d^{2}}{d x^{2}} (2 x^{3})|}_{1}, {\frac{d^{3}}{d x^{3}} (2 x^{3})|}_{- 2}$