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Section 3.12 : Higher Order Derivatives

8. Determine the second derivative of \(\displaystyle Q\left( v \right) = \frac{2}{{{{\left( {6 + 2v - {v^2}} \right)}^4}}}\)

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Not much to this problem other than to take two derivatives so each step will show each successive derivative until we get to the second. We’ll do a quick rewrite of the function to help with the derivatives and then the first derivative is,

\[\begin{align*}Q\left( v \right) &= 2{\left( {6 + 2v - {v^2}} \right)^{ - \,4}}\\ Q'\left( v \right) & = - 8\left( {2 - 2v} \right){\left( {6 + 2v - {v^2}} \right)^{ - \,5}}\end{align*}\] Show Step 2

Do not forget that often we will end up needing to do a product rule in the second derivative even though we did not need to do that in the first derivative. The second derivative is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{Q''\left( v \right) = 16{{\left( {6 + 2v - {v^2}} \right)}^{ - \,5}} + 40{{\left( {2 - 2v} \right)}^2}{{\left( {6 + 2v - {v^2}} \right)}^{ - \,6}}}}\]