Identify the function h(t) that needs to be differentiated. The function is given as h(t) = \(\frac{1}{2}\)t^3 + \(\frac{4}{t^2}\) + 3\(\sqrt{t}\).
Rewrite the function in a form that is easier to differentiate. This involves expressing each term with exponents: h(t) = \(\frac{1}{2}\)t^3 + 4t^{-2} + 3t^{1/2}.
Apply the power rule to differentiate each term separately. The power rule states that if f(t) = t^n, then f'(t) = nt^{n-1}.
Differentiate the first term: \(\frac{1}{2}\)t^3 becomes \(\frac{3}{2}\)t^2.
Differentiate the second term: 4t^{-2} becomes -8t^{-3}, and the third term: 3t^{1/2} becomes \(\frac{3}{2}\)t^{-1/2}.