To solve for dy/dt, we need to find the derivative of the function y with respect to t.
The function, y, is the product of two separate functions of t: (t^2 + 7t + 5) and (7t^2 + 3). This means we will be using the product rule for taking derivatives. 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.
Let's denote (t^2 + 7t + 5) as f(t) and (7t^2 + 3) as g(t).
The derivative of f(t), written as f'(t), is the derivative of (t^2 + 7t + 5) with respect to t, which equals 2t + 7.
The derivative of g(t), written as g'(t), is the derivative of (7t^2 + 3) with respect to t, which equals 14t.
Now we can apply the product rule:
dy/dt = f'(t)g(t) + f(t)g'(t)
Substituting the functions and their derivatives, we get:
dy/dt = (2t + 7)(7t^2 + 3) + (t^2 + 7t + 5)(14t)
Simplify to get the final answer:
dy/dt = 14t*(t^2 + 7t + 5) + (2t + 7)*(7t^2 + 3)
And that's your final solution.