Find dy/dt.
3) $y=t^{4}\left(t^{5}+3\right)^{4}$

Step 1 ：We are given the function \(y=t^{4}(t^{5}+3)^{4}\) and we are asked to find \(\frac{dy}{dt}\).

Step 2 ：We will use the product rule and the chain rule to find the derivative. The product rule states that the derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 3 ：Let's first find the derivative of \(t^{4}\), which is \(4t^{3}\).

Step 4 ：Next, we find the derivative of \((t^{5}+3)^{4}\) using the chain rule. The outer function is \(u^{4}\) and the inner function is \(t^{5}+3\). The derivative of the outer function is \(4u^{3}\) and the derivative of the inner function is \(5t^{4}\). So, the derivative of \((t^{5}+3)^{4}\) is \(4(t^{5}+3)^{3} \cdot 5t^{4} = 20t^{4}(t^{5}+3)^{3}\).

Step 5 ：Applying the product rule, we get \(\frac{dy}{dt} = 4t^{3}(t^{5}+3)^{4} + t^{4} \cdot 20t^{4}(t^{5}+3)^{3} = 4t^{3}(t^{5}+3)^{4} + 20t^{8}(t^{5}+3)^{3}\).

Step 6 ：Thus, the derivative of the function \(y=t^{4}(t^{5}+3)^{4}\) with respect to time (t) is \(\boxed{20t^{8}(t^{5}+3)^{3} + 4t^{3}(t^{5}+3)^{4}}\).