Q. 52 Calculate each of the derivative... [FREE SOLUTION]

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Chapter 2: Q. 52 (page 210)

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d}{d x} {(\frac{x^{2} + 1}{(x^{2} + 4) (3 x - 2)})}_{x = - 1}$

Short Answer

Expert verified

$\frac{d}{d x} {(\frac{x^{2} + 1}{(x^{2} + 4) (3 x - 2)})}_{x = - 1}$ is $\frac{- 24}{625}$

Step by step solution

Step 1. Given Information

We need to find $\frac{d}{d x} {(\frac{x^{2} + 1}{(x^{2} + 4) (3 x - 2)})}_{x = - 1}$

Let us write $y = \frac{x^{2} + 1}{(x^{2} + 4) (3 x - 2)}$

Step 2. Finding First Derivative

$\frac{d y}{d x} = \frac{(x^{2} + 4) (3 x - 2) \frac{d}{d x} (x^{2} + 1) - (x^{2} + 1) \frac{d}{d x} (x^{2} + 4) (3 x - 2)}{{((x^{2} + 4) (3 x - 2))}^{2}} \frac{d y}{d x} = \frac{(x^{2} + 4) (3 x - 2) (2 x) - (x^{2} + 1) [2 x (3 x - 2) + (x^{2} + 4) 3]}{{(x^{2} + 4)}^{2} {(3 x - 2)}^{2}} \frac{d y}{d x} = \frac{(x^{2} + 4) (6 x^{2} - 4 x) - (x^{2} + 1) [6 x^{2} - 4 x + 3 x^{2} + 12]}{{(x^{2} + 4)}^{2} {(3 x - 2)}^{2}} \frac{d y}{d x} = \frac{(x^{2} + 4) (6 x^{2} - 4 x) - (x^{2} + 1) (9 x^{2} - 4 x + 12)}{{(x^{2} + 4)}^{2} {(3 x - 2)}^{2}} = \frac{- 3 x^{4} + 3 x^{2} + 12 x - 12}{{(x^{2} + 4)}^{2} {(3 x - 2)}^{2}}$

Step 3. Finding derivative at x=-1

$\frac{d y}{d x} = \frac{- 3 {(- 1)}^{4} + 3 {(- 1)}^{2} + 12 (- 1) - 12}{{(1 + 4)}^{2} {(- 3 - 2)}^{2}} \frac{d y}{d x} = \frac{- 3 + 3 - 12 - 12}{{(1 + 4)}^{2} {(- 3 - 2)}^{2}} = \frac{- 24}{625}$

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91影视

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d}{d x} {(\frac{x^{2} + 1}{(x^{2} + 4) (3 x - 2)})}_{x = - 1}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First Derivative

Step 3. Finding derivative at x=-1

One App. One Place for Learning.

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91影视

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.ddxx2+1x2+43x-2x=-1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding First Derivative

Step 3. Finding derivative at x=-1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Pure Maths

Probability and Statistics

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Calculate each of the derivatives or derivative values in Exercises 47鈥�52.
$\frac{d}{d x} {(\frac{x^{2} + 1}{(x^{2} + 4) (3 x - 2)})}_{x = - 1}$