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: 26

Answer

\[{y^,} = - 5\ln 4\,\left( {{4^{ - 5x + 2}}} \right)\]

Work Step by Step

\[\begin{gathered} y = {4^{ - 5x + 2}} \hfill \\ Find\,\,the\,\,derivative \hfill \\ {y^,} = \,{\left( {{4^{ - 5x + 2}}} \right)^,} \hfill \\ Use\,\,the\,\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {{a^{g\,\left( x \right)}}} \right] = \,\left( {\ln a} \right){a^{g\,\left( x \right)}}{g^,}\,\left( x \right) \hfill \\ Then \hfill \\ Let\,\,a = 4\,,\,g\,\left( x \right) = - 5x + 2 \hfill \\ {y^,} = \,\left( {\ln 4} \right){4^{ - 5x + 2}}\,{\left( {5x + 2} \right)^,} \hfill \\ {y^,} = \,\left( {\ln 4} \right)\,\left( {{4^{ - 5x + 2}}} \right)\,\left( { - 5} \right) \hfill \\ Multiplying \hfill \\ {y^,} = - 5\ln 4\,\left( {{4^{ - 5x + 2}}} \right) \hfill \\ \end{gathered} \]
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