Chapter 2: Q. 48 (page 210)

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

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

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

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First Order Derivative

Step 3. Finding second order derivative

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Calculate each of the derivatives or derivative values in Exercises 47–52.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

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

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