Step 1:Given

\(\displaystyle{f{{\left({x},{y}\right)}}}={\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}}}\)

Step 2:Solution

\(\displaystyle{f{{\left({x},{y}\right)}}}={\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}}}\)

\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={5}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}-{1}}}{\left({3}{y}\right)}={15}{y}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}{{f}_{{{y}}}{\left({x},{y}\right)}}\)

\(\displaystyle={5}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}-{1}}}{\left({3}{x}+{4}{\left({2}\right)}{y}^{{{2}-{1}}}\right)}\)

\(\displaystyle={5}{\left({3}{x}+{8}{y}\right)}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)

Answer:

\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={15}{y}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)

\(\displaystyle{{f}_{{{y}}}{\left({x},{y}\right)}}={5}{\left({3}{x}+{8}{y}\right)}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)

\(\displaystyle{f{{\left({x},{y}\right)}}}={\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}}}\)

Step 2:Solution

\(\displaystyle{f{{\left({x},{y}\right)}}}={\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}}}\)

\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={5}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}-{1}}}{\left({3}{y}\right)}={15}{y}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}{{f}_{{{y}}}{\left({x},{y}\right)}}\)

\(\displaystyle={5}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}-{1}}}{\left({3}{x}+{4}{\left({2}\right)}{y}^{{{2}-{1}}}\right)}\)

\(\displaystyle={5}{\left({3}{x}+{8}{y}\right)}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)

Answer:

\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={15}{y}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)

\(\displaystyle{{f}_{{{y}}}{\left({x},{y}\right)}}={5}{\left({3}{x}+{8}{y}\right)}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)