Find the directional derivative of f at the given point

Michael Maggard 2022-01-04 Answered
Find the directional derivative of f at the given point in the direction indicated by the angle theta.
\(\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{-{x}}}\), \(\displaystyle{\left({0},{4}\right)}\), \(\displaystyle\theta={\frac{{{2}\pi}}{{{3}}}}\)

Expert Community at Your Service

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

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Lindsey Gamble
Answered 2022-01-05 Author has 4706 answers
\(\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{-{x}}}\Rightarrow{{f}_{{x}}{\left({x},{y}\right)}}=-{y}{e}^{{-{x}}}\) and \(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}={e}^{{-{x}}}\). If u is a unit vector in the direction \(\displaystyle\theta={\frac{{{2}\pi}}{{{3}}}}\), then
\(\displaystyle{D}_{{u}}{f{{\left({0},{4}\right)}}}={{f}_{{x}}{\left({0},{4}\right)}}{\cos{{\left({\frac{{{2}\pi}}{{{3}}}}\right)}}}+{{f}_{{y}}{\left({0},{4}\right)}}{\sin{{\left({\frac{{{2}\pi}}{{{3}}}}\right)}}}=-{4}\times{\frac{{-{1}}}{{{2}}}}+{1}\times{\frac{{\sqrt{{{3}}}}}{{{2}}}}={2}+{\frac{{\sqrt{{{3}}}}}{{{2}}}}\)
Not exactly what you’re looking for?
Ask My Question
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 2021-09-08
Find the directional derivative of f at the given point in the direction indicated by the angle \(\displaystyle\theta.{f{{\left({x},{y}\right)}}}={e}^{{x}}{\cos{{y}}},{\left({0},{0}\right)},\theta={\frac{{\pi}}{{{4}}}}\)
asked 2021-09-18
Find the directional derivative of f at the given point in the direction indicated by the angle theta. \(\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{-{x}}},{\left({0},{4}\right)},\theta={\frac{{{2}\pi}}{{{3}}}}\)
asked 2021-05-19
Find the directional derivative of f at the given point in the direction indicated by the angle \(\theta. f(x,y)=e^x\cos y,(0,0),\theta=\frac{\pi}{4}\)
asked 2021-05-29
Find the directional derivative of f at the given point in the direction indicated by the angle theta. \(f(x,y)=ye^{-x}, (0,4), \theta=\frac{2\pi}{3}\)
asked 2021-09-15
Find the maximum rate of change off at the given point and the direction in which it occurs.
\(\displaystyle{f{{\left({x},{y}\right)}}}={4}{y}\sqrt{{{x}}},{\left({4},{1}\right)}\)
asked 2021-05-28
Find the maximum rate of change off at the given point and the direction in which it occurs.
\(f(x, y) = 4y \sqrt{x}, (4,1)\)
asked 2021-09-23
Find the directional derivative of the function at the given point in the direction of the vector v. \(\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{x}}{{x}^{{2}}}+{y}^{{2}},{\left({1},{2}\right)},{u}={<}{3},{5}{>}\)
...