Question

# Use the Differentiation Formulas and Rules of Derivatives to find the derivatives of the following functions. h(y)=2y^{-4}-7y^{-3}+4y^{-2}+\frac{11}{y} h'(y)=

Derivatives
Use the Differentiation Formulas and Rules of Derivatives to find the derivatives of the following functions.
$$\displaystyle{h}{\left({y}\right)}={2}{y}^{{-{4}}}-{7}{y}^{{-{3}}}+{4}{y}^{{-{2}}}+{\frac{{{11}}}{{{y}}}}$$
$$\displaystyle{h}'{\left({y}\right)}=$$

2021-05-18
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 $$\displaystyle{h}'{\left({y}\right)}=-{\frac{{{8}}}{{{y}^{{{5}}}}}}+{\frac{{{21}}}{{{y}^{{{4}}}}}}-{\frac{{{8}}}{{{y}^{{{3}}}}}}-{\frac{{{11}}}{{{y}^{{{2}}}}}}$$.