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Section 3.12 : Higher Order Derivatives
4. Determine the fourth derivative of \(\displaystyle f\left( w \right) = 7\sin \left( \frac{w}{3} \right) + \cos \left( {1 - 2w} \right)\)
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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. The first derivative is then,
\[f'\left( w \right) = \frac{7}{3}\cos \left( \frac{w}{3}\right) + 2\sin \left( {1 - 2w} \right)\] Show Step 2The second derivative is,
\[f''\left( w \right) = - \frac{7}{9}\sin \left( \frac{w}{3} \right) - 4\cos \left( {1 - 2w} \right)\] Show Step 3The third derivative is,
\[f'''\left( w \right) = - \frac{7}{{27}}\cos \left(\frac{w}{3}\right) - 8\sin \left( {1 - 2w} \right)\] Show Step 4The fourth, and final derivative for this problem, is,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{{f^{\left( 4 \right)}}\left( w \right) = \frac{7}{{81}}\sin \left( \frac{w}{3}\right) + 16\cos \left( {1 - 2w} \right)}}\]