Find the first partial derivatives of the following functions. f(x,y)=(3xy+4y^{2}+1)^{5}

Tabansi 2021-05-28 Answered
Find the first partial derivatives of the following functions.
\(\displaystyle{f{{\left({x},{y}\right)}}}={\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{5}}}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

joshyoung05M
Answered 2021-05-29 Author has 16996 answers
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:
\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={15}{y}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)
\(\displaystyle{{f}_{{{y}}}{\left({x},{y}\right)}}={5}{\left({3}{x}+{8}{y}\right)}{\left({3}{x}{y}+{4}{y}^{{{2}}}+{1}\right)}^{{{4}}}\)
Have a similar question?
Ask An Expert
36
 
content_user
Answered 2021-11-20 Author has 2083 answers

Answer is given below (on video)

0

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more
...