Q. 55 Use the definition of the deriva... [FREE SOLUTION]

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Chapter 2: Q. 55 (page 185)

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

Short Answer

Expert verified

The derivatives are $6 x^{2}, 12 x, 12$ 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

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

Calculating the other derivatives

Calculating, we get

$\lim_{h 鈫� 0} \frac{g (x + h) - g (x)}{h} = \lim_{h 鈫� 0} \frac{6 (x + h)^{2} - 6 x^{2}}{h} \lim_{h 鈫� 0} \frac{g (x + h) - g (x)}{h} = \lim_{h 鈫� 0} \frac{6 (x^{2} + h^{2} + 2 x h) - 6 x^{2}}{h} = \lim_{h 鈫� 0} \frac{6 x^{2} + 6 h^{2} + 12 x h - 6 x^{2}}{h} = \lim_{h 鈫� 0} \frac{6 h^{2} + 12 x h}{h} = \lim_{h 鈫� 0} (6 h + 12 x) = 12 x \frac{d^{2}}{d x^{2}} (2 x^{3}) = 12 x$

Calculating the other derivatives

Calculating, we get

$\lim_{h 鈫� 0} \frac{m (x + h) - m (x)}{h} = \lim_{h 鈫� 0} \frac{12 (x + h) - 12 x}{h} \lim_{h 鈫� 0} \frac{m (x + h) - m (x)}{h} = \lim_{h 鈫� 0} \frac{12 x + 12 h - 12 x}{h} = \lim_{h 鈫� 0} \frac{12 h}{h} = \lim_{h 鈫� 0} 12 = 12 \frac{d^{3}}{d x^{3}} (2 x^{3}) = 12$

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91影视

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

Short Answer

Step by step solution

Step 1. Given information

Use the definition of derivative and calculate

Calculating the other derivatives

Calculating the other derivatives

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91影视

Use the definition of the derivative to find the derivatives described in Exercises 55-58.Findddx2x3,d2dx22x3,d3dx32x3

Short Answer

Step by step solution

Step 1. Given information

Use the definition of derivative and calculate

Calculating the other derivatives

Calculating the other derivatives

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Geometry

Decision Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

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