Step 1: Consider the given

Function, \(\displaystyle{h}{\left({y}\right)}={2}{y}^{{-{4}}}-{7}{y}^{{-{3}}}+{4}{y}^{{-{2}}}+{\frac{{{11}}}{{{y}}}}\)

Step 2: The objective

Is to find the derivative of the given function.

Step 3: Calculation

We will use the rule, \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{n}}}\right)}={n}{x}^{{{n}-{1}}}\) to find the derivative as follows,

\(\displaystyle{h}'{\left({y}\right)}={2}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{4}}}\right)}-{7}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{3}}}\right)}+{4}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{2}}}\right)}+{11}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{1}}}\right)}\)

\(\displaystyle={2}{\left(-{4}\right)}{y}^{{-{4}-{1}}}-{7}{\left(-{3}\right)}{y}^{{-{3}-{1}}}+{4}{\left(-{2}\right)}{y}^{{-{2}-{1}}}+{11}{\left(-{1}\right)}{y}^{{-{1}-{1}}}\)

\(\displaystyle=-{8}{y}^{{-{5}}}+{21}{y}^{{-{4}}}-{8}{y}^{{-{3}}}-{11}{y}^{{-{2}}}\)

\(\displaystyle={\frac{{-{8}}}{{{y}^{{{5}}}}}}+{\frac{{{21}}}{{{y}^{{{4}}}}}}-{\frac{{{8}}}{{{y}^{{{3}}}}}}-{\frac{{{11}}}{{{y}^{{{2}}}}}}\)

Step 4: Conclusion

Hence, we can conclude that \(\displaystyle{h}'{\left({y}\right)}=-{\frac{{{8}}}{{{y}^{{{5}}}}}}+{\frac{{{21}}}{{{y}^{{{4}}}}}}-{\frac{{{8}}}{{{y}^{{{3}}}}}}-{\frac{{{11}}}{{{y}^{{{2}}}}}}\).

Function, \(\displaystyle{h}{\left({y}\right)}={2}{y}^{{-{4}}}-{7}{y}^{{-{3}}}+{4}{y}^{{-{2}}}+{\frac{{{11}}}{{{y}}}}\)

Step 2: The objective

Is to find the derivative of the given function.

Step 3: Calculation

We will use the rule, \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{n}}}\right)}={n}{x}^{{{n}-{1}}}\) to find the derivative as follows,

\(\displaystyle{h}'{\left({y}\right)}={2}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{4}}}\right)}-{7}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{3}}}\right)}+{4}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{2}}}\right)}+{11}{\frac{{{d}}}{{{\left.{d}{y}\right.}}}}{\left({y}^{{-{1}}}\right)}\)

\(\displaystyle={2}{\left(-{4}\right)}{y}^{{-{4}-{1}}}-{7}{\left(-{3}\right)}{y}^{{-{3}-{1}}}+{4}{\left(-{2}\right)}{y}^{{-{2}-{1}}}+{11}{\left(-{1}\right)}{y}^{{-{1}-{1}}}\)

\(\displaystyle=-{8}{y}^{{-{5}}}+{21}{y}^{{-{4}}}-{8}{y}^{{-{3}}}-{11}{y}^{{-{2}}}\)

\(\displaystyle={\frac{{-{8}}}{{{y}^{{{5}}}}}}+{\frac{{{21}}}{{{y}^{{{4}}}}}}-{\frac{{{8}}}{{{y}^{{{3}}}}}}-{\frac{{{11}}}{{{y}^{{{2}}}}}}\)

Step 4: Conclusion

Hence, we can conclude that \(\displaystyle{h}'{\left({y}\right)}=-{\frac{{{8}}}{{{y}^{{{5}}}}}}+{\frac{{{21}}}{{{y}^{{{4}}}}}}-{\frac{{{8}}}{{{y}^{{{3}}}}}}-{\frac{{{11}}}{{{y}^{{{2}}}}}}\).