Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 4 - Calculating the Derivative - 4.4 Derivatives of Exponential Functions - 4.4 Exercises: 30

Answer

\[{s^,} = \frac{{5\ln 2\,\left( {{2^{\sqrt {t - 2} }}} \right)}}{{2\sqrt {t - 2} }}\]

Work Step by Step

\[\begin{gathered} s = 5 \cdot {2^{\sqrt {t - 2} }} \hfill \\ Find\,\,the\,\,derivative \hfill \\ {s^,} = \,\,\,{\left[ {5 \cdot {2^{\sqrt {t - 2} }}} \right]^,} \hfill \\ Pull\,\,out\,\,the\,\,constant\,\,5 \hfill \\ {s^,} = 5\,\,{\left[ {{2^{\sqrt {t - 2} }}} \right]^,} \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {{a^{g\,\left( x \right)}}} \right] = \,\left( {\ln a} \right){a^{g\,\left( x \right)}}{g^,}\,\left( x \right) \hfill \\ Then \hfill \\ {s^,} = 5\,\left( {\ln 2} \right)\,\left( {{2^{\sqrt {t - 2} }}} \right)\,{\left( {\sqrt {t - 2} } \right)^,} \hfill \\ differentiating\,\,\sqrt {t - 2} \hfill \\ {s^,} = 5\,\left( {\ln 2} \right)\,\left( {{2^{\sqrt {t - 2} }}} \right)\,\,\left( {\frac{1}{{2\sqrt {t - 2} }}} \right) \hfill \\ Multiply \hfill \\ {s^,} = \frac{{5\ln 2\,\left( {{2^{\sqrt {t - 2} }}} \right)}}{{2\sqrt {t - 2} }} \hfill \\ \end{gathered} \]
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