Step 1 Derivative

Use the rules for derivatives to find the derivative of function defined as follows.

\(\displaystyle{q}={\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{4}}}\)

Step 2 Formulation

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{n}}}\right)}={n}{x}^{{{n}-{1}}}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{x}}}\right)}={e}^{{{x}}}\)

Step 3 Solution

\(\displaystyle{q}={\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{4}}}\)

\(\displaystyle{\frac{{{d}{q}}}{{{d}{p}}}}={4}{\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{3}}}\cdot{e}^{{{2}{p}+{1}}}\cdot{2}\)

\(\displaystyle{\frac{{{d}{q}}}{{{d}{p}}}}={8}{\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{3}}}\cdot{e}^{{{2}{p}+{1}}}\)

Use the rules for derivatives to find the derivative of function defined as follows.

\(\displaystyle{q}={\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{4}}}\)

Step 2 Formulation

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{n}}}\right)}={n}{x}^{{{n}-{1}}}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{x}}}\right)}={e}^{{{x}}}\)

Step 3 Solution

\(\displaystyle{q}={\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{4}}}\)

\(\displaystyle{\frac{{{d}{q}}}{{{d}{p}}}}={4}{\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{3}}}\cdot{e}^{{{2}{p}+{1}}}\cdot{2}\)

\(\displaystyle{\frac{{{d}{q}}}{{{d}{p}}}}={8}{\left({e}^{{{2}{p}+{1}}}-{2}\right)}^{{{3}}}\cdot{e}^{{{2}{p}+{1}}}\)