Final answer:
To find the derivative of the given function f(t) = (1/2)(8t² + t), we can use the product rule. The derivative is f'(t) = 8t + 1/2.
Step-by-step explanation:
To find the derivative of the given function, we can use the product rule. The product rule states that if we have a function of the form f(t) = g(t) * h(t), then its derivative is given by f'(t) = g'(t) * h(t) + g(t) * h'(t).
Applying the product rule to the given function f(t) = (1/2)(8t^2 + t), we have:
f'(t) = (1/2)' * (8t^2 + t) + (1/2) * (8t^2 + t)'
Since the derivative of a constant (in this case 1/2) is 0, the first term simplifies to 0. Now, we need to find the derivative of the second term, which is (8t^2 + t)'
To find the derivative of (8t^2 + t), we use the power rule for differentiation. The power rule states that if we have a term of the form t^n, then its derivative is given by (t^n)' = n * t^(n-1).
Applying the power rule to the term 8t^2, we have:
(8t^2)' = 2 * 8t^(2-1) = 16t.
Applying the power rule to the term t, we have:
(t)' = 1 * t^(1-1) = 1.
Therefore, the derivative of (8t^2 + t) is 16t + 1.
Substituting this result back into the derivative of f(t), we have:
f'(t) = 0 + (1/2) * (16t + 1)
Simplifying, we get:
f'(t) = 8t + 1/2