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Q. 28

Expert-verified

Calculus

Found in: Page 210

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.
$f (x) = \frac{{(1 + \sqrt{x})}^{2}}{3 x^{2} - 4 x + 1}$

The required answer is $\frac{1 + \sqrt{x}}{\sqrt{x} (3 x^{2} - 4 x + 1)} - \frac{{(1 + \sqrt{x})}^{2} (6 x - 4)}{{(3 x^{2} - 4 x + 1)}^{2}}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

The given function is $f (x) = \frac{{(1 + \sqrt{x})}^{2}}{3 x^{2} - 4 x + 1}$

Step 2. Calculation

Differentiate both the sides with respect to x, we get,

$f' (x) = \frac{(\frac{d}{d x} {(1 + \sqrt{x})}^{2}) (3 x^{2} - 4 x + 1) - {(1 + \sqrt{x})}^{2} (\frac{d}{d x} (3 x^{2} - 4 x + 1))}{{(3 x^{2} - 4 x + 1)}^{2}} = \frac{(2 (1 + \sqrt{x}) \frac{d}{d x} (1 + \sqrt{x})) (3 x^{2} - 4 x + 1) - {(1 + \sqrt{x})}^{2} (3 (2 x) - 4)}{{(3 x^{2} - 4 x + 1)}^{2}} = \frac{(2 (1 + \sqrt{x}) \frac{1}{2} x^{- \frac{1}{2}}) (3 x^{2} - 4 x + 1) - {(1 + \sqrt{x})}^{2} (6 x) - 4)}{{(3 x^{2} - 4 x + 1)}^{2}} = \frac{(\frac{1 + \sqrt{x}}{\sqrt{x}}) (3 x^{2} - 4 x + 1) - {(1 + \sqrt{x})}^{2} (6 x) - 4)}{{(3 x^{2} - 4 x + 1)}^{2}} = \frac{(1 + \sqrt{x})}{\sqrt{x} {(3 x^{2} - 4 x + 1)}^{2}} - \frac{{(1 + \sqrt{x})}^{2} (6 x) - 4)}{{(3 x^{2} - 4 x + 1)}^{2}}$

Most popular questions for Math Textbooks

State the chain rule for differentiating a composition $g (h (x))$ of two functions expressed

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Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park $t$ minutes after she begins her jog is given by the function $s (t)$ shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

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$f (x) = x^{3} + 1, [a, b] = [- 2, 1]$

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$f (x) = {(1 - 4 x)}^{2} {(3 x^{2} + 1)}^{9}$

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