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

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

Calculating the other derivatives

Calculating, we get

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

Calculating the other derivatives

Calculating, we get

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

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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

One App. One Place for Learning.

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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

Probability and Statistics

Applied Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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