Q. 48 Calculate each of the derivative... [FREE SOLUTION]

Chapter 2: Q. 48 (page 210)

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d^{2}}{d x^{2}} (\frac{\frac{1}{x} - 3 x^{2}}{x^{5} - \frac{1}{\sqrt{x}}})$

Short Answer

Expert verified

The second derivative of $\frac{d^{2}}{d x^{2}} (\frac{\frac{1}{x} - 3 x^{2}}{x^{5} - \frac{1}{\sqrt{x}}})$ is $\frac{- 393 x^{\frac{29}{2}} - 97 x^{\frac{23}{2}} + 45 x^{\frac{7}{2}} - 3 \sqrt{x} - 144 x^{20} + 168 x^{17} + 492 x^{9} - 68 x^{6}}{- 16 x^{\frac{39}{2}} - 16 x^{\frac{17}{2}} + 4 x^{25} + 24 x^{14} + 4 x^{3}}$

Step by step solution

Step 1. Given Information

We need to ask second order derivative of $\frac{d^{2}}{d x^{2}} (\frac{\frac{1}{x} - 3 x^{2}}{x^{5} - \frac{1}{\sqrt{x}}})$

Let us write $y = (\frac{\frac{1}{x} - 3 x^{2}}{x^{5} - \frac{1}{\sqrt{x}}})$

Step 2. Finding First Order Derivative

$\frac{d}{d x} (\frac{- 3 x^{2} + \frac{1}{x}}{x^{5} - \frac{1}{\sqrt{x}}}) = (\frac{\frac{d}{d x} (- 3 x^{2} + \frac{1}{x}) (x^{5} - \frac{1}{\sqrt{x}}) - (- 3 x^{2} + \frac{1}{x}) \frac{d}{d x} (x^{5} - \frac{1}{\sqrt{x}})}{{(x^{5} - \frac{1}{\sqrt{x}})}^{2}}) = (\frac{(- 6 x - \frac{1}{x^{2}}) (x^{5} - \frac{1}{\sqrt{x}}) - (- 3 x^{2} + \frac{1}{x}) (5 x^{4} + \frac{1}{2 x^{\frac{3}{2}}})}{{(x^{5} - \frac{1}{\sqrt{x}})}^{2}}) = \frac{- 18 x^{\frac{17}{2}} + 12 x^{\frac{11}{2}} - 15 x^{3} - 1}{- 2 x^{\frac{25}{2}} - 2 x^{\frac{3}{2}} + 4 x^{7}}$

Step 3. Finding second order derivative

$\frac{d^{2} y}{d x^{2}} = \frac{(- 2 x^{\frac{25}{2}} - 2 x^{\frac{3}{2}} + 4 x^{7}) \frac{d}{d x} (- 18 x^{\frac{17}{2}} + 12 x^{\frac{11}{2}} - 15 x^{3} - 1) - (- 18 x^{\frac{17}{2}} + 12 x^{\frac{11}{2}} - 15 x^{3} - 1) \frac{d}{d x} (- 2 x^{\frac{25}{2}} - 2 x^{\frac{3}{2}} + 4 x^{7})}{{(- 2 x^{\frac{25}{2}} - 2 x^{\frac{3}{2}} + 4 x^{7})}^{2}} \frac{d^{2} y}{d x^{2}} = \frac{(- 2 x^{\frac{25}{2}} - 2 x^{\frac{3}{2}} + 4 x^{7}) (- 153 x^{15 / 2} + 66 x^{\frac{9}{2}} - 45 x^{2} - 0) - (- 18 x^{\frac{17}{2}} + 12 x^{\frac{11}{2}} - 15 x^{3} - 1) (- 25 x^{\frac{23}{2}} - 3 \sqrt{x} + 28 x^{6})}{{(- 2 x^{\frac{25}{2}} - 2 x^{\frac{3}{2}} + 4 x^{7})}^{2}} \frac{d^{2} y}{d x^{2}} = \frac{- 393 x^{\frac{29}{2}} - 97 x^{\frac{23}{2}} + 45 x^{\frac{7}{2}} - 3 \sqrt{x} - 144 x^{20} + 168 x^{17} + 492 x^{9} - 68 x^{6}}{- 16 x^{\frac{39}{2}} - 16 x^{\frac{17}{2}} + 4 x^{25} + 24 x^{14} + 4 x^{3}}$

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

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d^{2}}{d x^{2}} (\frac{\frac{1}{x} - 3 x^{2}}{x^{5} - \frac{1}{\sqrt{x}}})$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First Order Derivative

Step 3. Finding second order derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Geometry

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.d2dx21x-3x2x5-1x

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First Order Derivative

Step 3. Finding second order derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Geometry

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d^{2}}{d x^{2}} (\frac{\frac{1}{x} - 3 x^{2}}{x^{5} - \frac{1}{\sqrt{x}}})$