If $f(t)=\left(t^{2}+3 t+7\right)\left(2 t^{2}+5\right)$, find $f^{\prime}(t)$ Find $f^{\prime}(3)$.

Step 1 ：Given the function \(f(t) = (t^{2}+3t+7)(2t^{2}+5)\)

Step 2 ：We need to find the derivative of this function, \(f'(t)\), using the product rule of differentiation.

Step 3 ：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 4 ：Applying the product rule, we get \(f'(t) = 4t(t^{2}+3t+7) + (2t+3)(2t^{2}+5)\)

Step 5 ：Now, we need to find \(f'(3)\). Substituting \(t = 3\) into \(f'(t)\), we get \(f'(3) = 507\)

Step 6 ：So, the final answer is \(\boxed{507}\)