Find all first-order partial derivatives of the function at any

prelimaf1 2021-11-20 Answered
Find all first-order partial derivatives of the function at any point of the domain.
\(\displaystyle{f{{\left({x},{y},{z}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{z}^{{{3}}}}}{{{x}^{{{2}}}}}}+{4}{y}-{x}^{{{16}}}+{2021}\)

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

Jeffrey Parrish
Answered 2021-11-21 Author has 151 answers
Step 1
To find all first - order partial derivatives of the function :-
\(\displaystyle{f{{\left({x},{y},{z}\right)}}}={4}{x}^{{{3}}}{y}^{{{2}}}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{z}^{{{3}}}}}{{{x}^{{{2}}}}}}+{4}{y}-{x}^{{{16}}}+{2021}\)
\(\displaystyle{f}_{{{x}}}={12}{x}^{{{2}}}{y}^{{{2}}}-{0}-{\frac{{{2}{z}^{{{3}}}}}{{{x}^{{{3}}}}}}+{0}-{16}{x}^{{{15}}}+{0}\)
\(\displaystyle{f}_{{{x}}}={12}{x}^{{{2}}}{y}^{{{2}}}-{\frac{{{2}{z}^{{{3}}}}}{{{x}^{{{3}}}}}}-{16}{x}^{{{15}}}\)
Step 2
\(\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{0}+{4}{\left({1}\right)}-{0}\)
\(\displaystyle{f}_{{{y}}}={8}{x}^{{{3}}}{y}-{4}{e}^{{{z}}}{y}^{{{3}}}+{4}\)
\(\displaystyle{f}_{{{z}}}={0}-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}+{0}\)
\(\displaystyle{f}_{{{z}}}=-{e}^{{{z}}}{y}^{{{4}}}+{\frac{{{3}{z}^{{{2}}}}}{{{x}^{{{2}}}}}}\)
Have a similar question?
Ask An Expert
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

Relevant Questions

asked 2021-07-05
Write first-order differential equations that express the situation: The increase in the number of people who have heard a rumor in a town of population 20,00020,000 is proportional to the product of the number PP who have heard the rumor and the number who have not heard the rumor at time tt .
asked 2021-05-03
How do you solve linear first-order differential equations?
asked 2021-05-26
Write the following first-order differential equations in standard form. \(\displaystyle{y}'={x}^{{{3}}}{y}+{\sin{{x}}}\)
asked 2021-05-22
Write the following first-order differential equations in standard form. \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={y}{x}{\left({x}+{1}\right)}\)
asked 2021-06-10
Write first-order differential equations that express the situation: A student in an engineering course finds the rate of increase of his grade point average GG is directly proportional to the number NN of study hours/week and inversely proportional to the amount AA of time spent on online games .
asked 2021-11-10
Find all of the first order partial derivatives for the following functions:
\(\displaystyle{v}={\ln{{\left({x}{\sin{{y}}}\right)}}}+{e}^{{{x}+{y}}}\)
asked 2021-10-15
Find all of the first order partial derivatives
\(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{{4}}}+{6}\sqrt{{{y}}}-{10}\)

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
...