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(1 point) Let $f(t)=\left(t^{2}+5 t+6\right)\left(2 t^{2}+6\right)$. Find $f^{\prime}(t)$.
\[
f^{\prime}(t)=
\]

Find $f^{\prime}(5)$.
\[
f^{\prime}(5)=
\]

Note: You can earn partial credit on this problem.
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Step 1 ：The problem is asking for the derivative of the function \(f(t)\) and then to evaluate this derivative at \(t=5\). To solve this, we first need to find the derivative of the function \(f(t)\) using the product rule. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 2 ：Applying the product rule, we find that the derivative of the function \(f(t)\) is \(f^{\prime}(t) = 4t(t^{2} + 5t + 6) + (2t + 5)(2t^{2} + 6)\).

Step 3 ：After finding the derivative, we substitute \(t=5\) into the derivative function to find \(f^{\prime}(5)\).

Step 4 ：Substituting \(t=5\) into the derivative function, we find that \(f^{\prime}(5) = 1960\).

Step 5 ：So, the final answers are \(f^{\prime}(t) = 4t(t^{2} + 5t + 6) + (2t + 5)(2t^{2} + 6)\) and \(f^{\prime}(5) = \boxed{1960}\).