Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 1 - Functions - 1.1 Review of Functions - 1.1 Exercises: 59

Answer

\[ = \frac{{ - 2}}{{\,\left( {x + h} \right)x}}\]

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = \frac{2}{x} \hfill \\ \hfill \\ Use{\text{ }}the{\text{ }}definition{\text{ }}of{\text{ }}derivative \hfill \\ \hfill \\ \frac{{f\,\left( {x + h} \right) - f\,\left( x \right)}}{h} = \frac{{\frac{2}{{x + h}} - \frac{2}{x}}}{h} \hfill \\ \hfill \\ combine\,\,fractions \hfill \\ \hfill \\ = \frac{{\frac{{2x - 2\,\left( {x + h} \right)}}{{\,\left( {x + h} \right)x}}}}{h} \hfill \\ \hfill \\ multiply \hfill \\ \hfill \\ = \frac{{\frac{{2x - 2x - 2h}}{{\,\left( {x + h} \right)x}}}}{h} \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \frac{{\frac{{ - 2h}}{{\,\left( {x + h} \right)x}}}}{h} \hfill \\ \hfill \\ = \frac{{ - 2h}}{{\,\left( {x + h} \right)xh}} \hfill \\ \hfill \\ = \frac{{ - 2}}{{\,\left( {x + h} \right)x}} \hfill \\ \end{gathered} \]
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