Find the indicated derivative.
$h\left(t\right)=\frac{1}{2}{t}^3+\frac{4}{t^2}+3\sqrt{t}$
$h^{\prime }\left(t\right)=$

• A. $t^2-\(\frac{2}{t^3}\)+\(\frac{1}{\sqrt{t}\)}$
• B. $\(\frac\)32t^2-8t^3+\(\frac\)32\(\sqrt{t}\)$
• C. $\frac{3}{2}{t}^2-\frac{8}{t^3}+\frac{3}{2\sqrt{t}}$
• D. $\frac{3}{2}{t}^3-\frac{8}{t^2}+\frac{3}{2}\sqrt{t}$

Answer: (C)

Verified step by step guidance

1. 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\(\sqrt{t}\).
2. 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}.
3. 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}.
4. Differentiate the first term: \(\frac{1}{2}\)t^3 becomes \(\frac{3}{2}\)t^2.
5. Differentiate the second term: 4t^{-2} becomes -8t^{-3}, and the third term: 3t^{1/2} becomes \(\frac{3}{2}\)t^{-1/2}.