Chapter 2: Q. 47 (page 210)

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

Short Answer

Expert verified

The second order derivatives of $\frac{d^{2}}{d x^{2}} ({(x \sqrt{x + 1})}^{- 2})$ islocalid="1649329655183" $\frac{2 (6 x^{2} + 8 x + 3)}{x^{4} {(x + 1)}^{3}}$

Step by step solution

Step 1. Given Information

We need to find $\frac{d^{2}}{d x^{2}} ({(x \sqrt{x + 1})}^{- 2})$

Let us write

$y = ({(x \sqrt{x + 1})}^{- 2}) y = \frac{1}{x^{2} {(\sqrt{x + 1})}^{2}} = \frac{1}{x^{2} (x + 1)}$

Step 2. Finding First Derivative

Differentiating with respect to $x$

localid="1649329091115" role="math" $\frac{d y}{d x} = \frac{0 x^{2} (x + 1) - [2 x (x + 1) - x^{2}]}{x^{4} {(x + 1)}^{2}} \frac{d y}{d x} = \frac{- 2 x^{2} - 2 x + x^{2}}{x^{4} {(x + 1)}^{2}} = \frac{- x^{2} - 2 x}{x^{4} {(x + 1)}^{2}}$

Step 3. Finding Second order Derivative

$\frac{d^{2} y}{d x^{2}} = \frac{(- 2 x - 2) (x^{4} (x + 1)) - (- x^{2} - 2 x) (4 x^{3} {(x + 1)}^{2} + x^{4} \times 2 (x + 1))}{x^{8} {(x + 1)}^{2}} \frac{d^{2} y}{d x^{2}} = \frac{(- 2 x - 2) (x^{5} + x^{4}) + (x^{2} + 2 x) (4 x^{5} + 8 x^{4} + 4 x^{3}) + 2 x^{5} + 2 x^{4}}{x^{8} {(x + 1)}^{2}} \frac{d^{2} y}{d x^{2}} = \frac{2 (6 x^{2} + 8 x + 3)}{x^{4} {(x + 1)}^{3}}$

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First Derivative

Step 3. Finding Second order Derivative

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Calculate each of the derivatives or derivative values in Exercises 47–52.d2dx2xx+1-2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First Derivative

Step 3. Finding Second order Derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Discrete Mathematics

Geometry

Calculus

Statistics

Study anywhere. Anytime. Across all devices.

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