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: 34

Answer

\[{f^,}\,\left( x \right) = {e^{\frac{{{x^2}}}{{{x^3} + 2}}}}\,\,\left( {\frac{{4x - {x^4}}}{{\,{{\left( {{x^3} + 2} \right)}^2}}}} \right)\]

Work Step by Step

\[\begin{gathered} f\,\left( x \right) = {e^{\frac{{{x^2}}}{{{x^3} + 2}}}} \hfill \\ Differentiate\,\,using\,\,the\,formula \hfill \\ \frac{d}{{dx}}\,\,\left[ {{e^{g\,\left( x \right)}}} \right] = {e^{g\,\left( x \right)}}{g^,}\,\left( x \right) \hfill \\ Then \hfill \\ {f^,}\,\left( x \right) = {e^{\frac{{{x^2}}}{{{x^3} + 2}}}}\,\,{\left[ {\frac{{{x^2}}}{{{x^3} + 2}}} \right]^,} \hfill \\ Use\,\,the\,\,quotient\,\,rule \hfill \\ {f^,}\,\left( x \right) = {e^{\frac{{{x^2}}}{{{x^3} + 2}}}}\,\,\left( {\frac{{\,\left( {{x^3} + 2} \right)\,{{\left( {{x^2}} \right)}^,} - {x^2}\,{{\left( {{x^3} + 2} \right)}^,}}}{{\,{{\left( {{x^3} + 2} \right)}^2}}}} \right) \hfill \\ Then \hfill \\ {f^,}\,\left( x \right) = {e^{\frac{{{x^2}}}{{{x^3} + 2}}}}\,\left( {\frac{{\left( {{x^3} + 2} \right)\,\left( {2x} \right) - {x^2}\,\left( {3{x^2}} \right)}}{{\,{{\left( {{x^3} + 2} \right)}^2}}}} \right) \hfill \\ Multiplying \hfill \\ \hfill \\ {f^,}\,\left( x \right) = {e^{\frac{{{x^2}}}{{{x^3} + 2}}}}\,\,\left( {\frac{{4x - {x^4}}}{{\,{{\left( {{x^3} + 2} \right)}^2}}}} \right) \hfill \\ \end{gathered} \]
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