Find the second derivative of the function.
6) $s=\frac{t^{8}+9 t+8}{t^{2}}$
Answer
\(\boxed{\text{Final Answer: The second derivative of the function } s=\frac{t^{8}+9 t+8}{t^{2}} \text{ is } \frac{54t^{8} - 18t - 16}{t^{4}} - 4\frac{t^{2}(8t^{7} + 9) - 2t(t^{8} + 9t + 8)}{t^{5}}.}\)
Step 1 ：Given the function \(s=\frac{t^{8}+9 t+8}{t^{2}}\), we are asked to find the second derivative.
Step 2 ：First, we need to find the first derivative. To do this, we use the quotient rule for differentiation, which states that the derivative of \(\frac{u}{v}\) is \(\frac{vu' - uv'}{v^2}\), where \(u\) and \(v\) are functions of \(t\), and \(u'\) and \(v'\) are their respective derivatives.
Step 3 ：In this case, \(u = t^{8}+9 t+8\) and \(v = t^{2}\).
Step 4 ：First, we find \(u'\) and \(v'\). \(u' = 8t^{7} + 9\) and \(v' = 2t\).
Step 5 ：Substituting these into the quotient rule, we find the first derivative: \(\frac{t^{2}(8t^{7} + 9) - 2t(t^{8} + 9t + 8)}{t^{4}}\).
Step 6 ：Next, we find the derivative of the first derivative to get the second derivative: \(\frac{54t^{8} - 18t - 16}{t^{4}} - 4\frac{t^{2}(8t^{7} + 9) - 2t(t^{8} + 9t + 8)}{t^{5}}\).
Step 7 ：\(\boxed{\text{Final Answer: The second derivative of the function } s=\frac{t^{8}+9 t+8}{t^{2}} \text{ is } \frac{54t^{8} - 18t - 16}{t^{4}} - 4\frac{t^{2}(8t^{7} + 9) - 2t(t^{8} + 9t + 8)}{t^{5}}.}\)
