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Section 3.12 : Higher Order Derivatives
3. Determine the fourth derivative of \(\displaystyle f\left( x \right) = 4\,\sqrt[5]{{{x^3}}} - \frac{1}{{8{x^2}}} - \sqrt x \)
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Start SolutionNot much to this problem other than to take four derivatives so each step will show each successive derivative until we get to the fourth. After a quick rewrite of the function to help with the differentiation the first derivative is,
\[f\left( x \right) = 4\,{x^{\frac{3}{5}}} - \frac{1}{8}{x^{ - 2}} - {x^{\frac{1}{2}}}\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}f'\left( x \right) = \frac{{12}}{5}\,{x^{ - \,\,\frac{2}{5}}} + \frac{1}{4}{x^{ - 3}} - \frac{1}{2}{x^{ - \,\,\frac{1}{2}}}\] Show Step 2The second derivative is,
\[f''\left( x \right) = - \frac{{24}}{{25}}\,{x^{ - \,\,\frac{7}{5}}} - \frac{3}{4}{x^{ - 4}} + \frac{1}{4}{x^{ - \,\,\frac{3}{2}}}\] Show Step 3The third derivative is,
\[f'''\left( x \right) = \frac{{168}}{{125}}\,{x^{ - \,\,\frac{{12}}{5}}} + 3{x^{ - 5}} - \frac{3}{8}{x^{ - \,\,\frac{5}{2}}}\] Show Step 4The fourth, and final derivative for this problem, is,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{{f^{\left( 4 \right)}}\left( x \right) = - \frac{{2016}}{{625}}\,{x^{ - \,\,\frac{{17}}{5}}} - 15{x^{ - 6}} + \frac{{15}}{{16}}{x^{ - \,\,\frac{7}{2}}}}}\]