Find the second derivative of the function.
6) $s = \frac{t^{8} + 9 t + 8}{t^{2}}$

Answer

Expert–verified

Hide Steps

Answer

$Final Answer: The second derivative of the function s = \frac{t^{8} + 9 t + 8}{t^{2}} is \frac{54 t^{8} - 18 t - 16}{t^{4}} - 4 \frac{t^{2} (8 t^{7} + 9) - 2 t (t^{8} + 9 t + 8)}{t^{5}} .$

Steps

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 $\frac{v u^{'} - u v^{'}}{v^{2}}$ , 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 $v^{'}$ . $u^{'} = 8 t^{7} + 9$ and $v^{'} = 2 t$ .

Step 5 ：Substituting these into the quotient rule, we find the first derivative: $\frac{t^{2} (8 t^{7} + 9) - 2 t (t^{8} + 9 t + 8)}{t^{4}}$ .

Step 6 ：Next, we find the derivative of the first derivative to get the second derivative: $\frac{54 t^{8} - 18 t - 16}{t^{4}} - 4 \frac{t^{2} (8 t^{7} + 9) - 2 t (t^{8} + 9 t + 8)}{t^{5}}$ .

Step 7 ： $Final Answer: The second derivative of the function s = \frac{t^{8} + 9 t + 8}{t^{2}} is \frac{54 t^{8} - 18 t - 16}{t^{4}} - 4 \frac{t^{2} (8 t^{7} + 9) - 2 t (t^{8} + 9 t + 8)}{t^{5}} .$

Find the second derivative of the function.6) s=t8+9t+8t2

Answer

Popular Problems

Find the second derivative of the function.
6) $s = \frac{t^{8} + 9 t + 8}{t^{2}}$