Find derivative of given function:

\(\displaystyle{s}'{\left({t}\right)}=^{{{1}}}{\left({6}{t}^{{-{2}}}\right)}'+{\left({3}{t}^{{{3}}}\right)}'-{\left({4}{t}^{{{\frac{{{1}}}{{{2}}}}}}\right)}'\)

\(\displaystyle=^{{{2}}}{6}\dot{{-{2}{t}^{{{3}}}}}+{3}\dot{{3}}{t}^{{{2}}}-{4}\dot{{\frac{{{1}}}{{{2}}}}}{t}^{{-{\frac{{{1}}}{{{2}}}}}}\)

\(\displaystyle=-{12}{t}^{{-{3}}}+{9}{t}^{{{2}}}-{2}{t}^{{-{\frac{{{1}}}{{{2}}}}}}\)

Rules used:

(1) Derivative of sum

(2) Power rule

\(\displaystyle{s}'{\left({t}\right)}=^{{{1}}}{\left({6}{t}^{{-{2}}}\right)}'+{\left({3}{t}^{{{3}}}\right)}'-{\left({4}{t}^{{{\frac{{{1}}}{{{2}}}}}}\right)}'\)

\(\displaystyle=^{{{2}}}{6}\dot{{-{2}{t}^{{{3}}}}}+{3}\dot{{3}}{t}^{{{2}}}-{4}\dot{{\frac{{{1}}}{{{2}}}}}{t}^{{-{\frac{{{1}}}{{{2}}}}}}\)

\(\displaystyle=-{12}{t}^{{-{3}}}+{9}{t}^{{{2}}}-{2}{t}^{{-{\frac{{{1}}}{{{2}}}}}}\)

Rules used:

(1) Derivative of sum

(2) Power rule