College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter R - Section R.7 - Rational Expressions - R.7 Assess Your Understanding - Page 71: 74


$-\displaystyle \frac{2x+h}{x^{2}(x+h)^{2}}$

Work Step by Step

First, simplify $\displaystyle \frac{1}{x+h}-\frac{1}{x}$ LCD = $x^{2}(x+h)^{2}$ $\displaystyle \frac{1}{(x+h)^{2}}\cdot\frac{x^{2}}{x^{2}}-\frac{1}{x^{2}}\cdot\frac{(x+h)^{2}}{(x+h)^{2}}=\frac{x^{2}-(x+h)^{2}}{x^{2}(x+h)^{2}}$ $=\displaystyle \frac{x^{2}-(x^{2}+2hx+h^{2})}{x^{2}(x+h)^{2}}$ $=\displaystyle \frac{-2hx-h^{2}}{x^{2}(x+h)^{2}}$ $=\displaystyle \frac{-h(2x+h)}{x^{2}(x+h)^{2}}$ Now, $\displaystyle \frac{1}{h}\left[\frac{1}{(x+h)^{2}}-\frac{1}{x^{2}}\right]$ = $\displaystyle \frac{1}{h}\left[\frac{-h(2x+h)}{x^{2}(x+h)^{2}}\right]$ $... $ h cancels $=\displaystyle \frac{-(2x+h)}{x^{2}(x+h)^{2}}$ $=-\displaystyle \frac{2x+h}{x^{2}(x+h)^{2}}$
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