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Gradient and directional derivatives practice problems

∇f, and the rate of change in any direction: D_u f = ∇f · u. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

Interactive: Gradient Explorer

18 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 10.14

hard Video
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{y}\\\frac{\partial}{\partial y} x e^{y}\end{matrix}\right] = \left[\begin{matrix}e^{y}\\x e^{y}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}e^{4}\\3 e^{4}\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ 2 \sqrt{2} e^{4} \]
    D_u f = ∇f · u.Reviewed
Answer \( 2 \sqrt{2} e^{4} \)
A contour map of f(x, y) = x·e^(y): each curve joins points where f has the same value. At (3, 4) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.17

hard Video
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y - y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y - y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} - 3 y^{2}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}4\\-11\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ - \frac{26 \sqrt{5}}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( - \frac{26 \sqrt{5}}{5} \)
A contour map of f(x, y) = x²·y - y³: each curve joins points where f has the same value. At (1, 2) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.82

hard Video
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{y}\\\frac{\partial}{\partial y} x e^{y}\end{matrix}\right] = \left[\begin{matrix}e^{y}\\x e^{y}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}e^{2}\\e^{2}\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ \frac{\sqrt{5} e^{2}}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( \frac{\sqrt{5} e^{2}}{5} \)
A contour map of f(x, y) = x·e^(y): each curve joins points where f has the same value. At (1, 2) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.213

hard Video
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y^{2} + 2 x\right)\\\frac{\partial}{\partial y} \left(x y^{2} + 2 x\right)\end{matrix}\right] = \left[\begin{matrix}y^{2} + 2\\2 x y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}3\\-2\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ - \frac{7 \sqrt{5}}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( - \frac{7 \sqrt{5}}{5} \)
A contour map of f(x, y) = x·y² + 2x: each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.214

hard Video
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} x e^{y}\\\frac{\partial}{\partial y} x e^{y}\end{matrix}\right] = \left[\begin{matrix}e^{y}\\x e^{y}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}e\\- e\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ - \frac{3 \sqrt{5} e}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( - \frac{3 \sqrt{5} e}{5} \)
A contour map of f(x, y) = x·e^(y): each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.246

hard Video
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y - y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y - y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} - 3 y^{2}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}4\\-11\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ - \frac{7 \sqrt{2}}{2} \]
    D_u f = ∇f · u.Reviewed
Answer \( - \frac{7 \sqrt{2}}{2} \)
A contour map of f(x, y) = x²·y - y³: each curve joins points where f has the same value. At (1, 2) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.281

hard Video
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + 3 x y + y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} + 3 x y + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x + 3 y\\3 x + 2 y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}1\\-1\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ - \frac{3 \sqrt{5}}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( - \frac{3 \sqrt{5}}{5} \)
A contour map of f(x, y) = x² + 3x·y + y²: each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.9

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ 0 \]
    D_u f = ∇f · u.Reviewed
Answer \( 0 \)
A contour map of f(x, y) = sqrt(x² + y²): each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.11

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}1\\0\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}1\\0\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ 1 \]
    D_u f = ∇f · u.Reviewed
Answer \( 1 \)
A contour map of f(x, y) = sqrt(x² + y²): each curve joins points where f has the same value. At (1, 0) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.12

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y^{2} + 2 x\right)\\\frac{\partial}{\partial y} \left(x y^{2} + 2 x\right)\end{matrix}\right] = \left[\begin{matrix}y^{2} + 2\\2 x y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}18\\24\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}1\\0\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ 18 \]
    D_u f = ∇f · u.Reviewed
Answer \( 18 \)
A contour map of f(x, y) = x·y² + 2x: each curve joins points where f has the same value. At (3, 4) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat ✓ Checked by a person Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.20

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + 3 x y + y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} + 3 x y + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x + 3 y\\3 x + 2 y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}2\\3\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ \frac{18}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( \frac{18}{5} \)
A contour map of f(x, y) = x² + 3x·y + y²: each curve joins points where f has the same value. At (1, 0) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.21

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + 3 x y + y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} + 3 x y + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x + 3 y\\3 x + 2 y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}1\\-1\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{5}{13}\\\frac{12}{13}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ - \frac{7}{13} \]
    D_u f = ∇f · u.Reviewed
Answer \( - \frac{7}{13} \)
A contour map of f(x, y) = x² + 3x·y + y²: each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.71

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} + 3 x y + y^{2}\right)\\\frac{\partial}{\partial y} \left(x^{2} + 3 x y + y^{2}\right)\end{matrix}\right] = \left[\begin{matrix}2 x + 3 y\\3 x + 2 y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}7\\8\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ \frac{9 \sqrt{5}}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( \frac{9 \sqrt{5}}{5} \)
A contour map of f(x, y) = x² + 3x·y + y²: each curve joins points where f has the same value. At (2, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.73

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y^{2} + 2 x\right)\\\frac{\partial}{\partial y} \left(x y^{2} + 2 x\right)\end{matrix}\right] = \left[\begin{matrix}y^{2} + 2\\2 x y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}3\\-2\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{3}{5}\\\frac{4}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( \frac{1}{5} \)
A contour map of f(x, y) = x·y² + 2x: each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.77

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}\frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ \frac{3}{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( \frac{3}{5} \)
A contour map of f(x, y) = sqrt(x² + y²): each curve joins points where f has the same value. At (1, 2) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.217

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{x^{2} + y^{2}}\\\frac{\partial}{\partial y} \sqrt{x^{2} + y^{2}}\end{matrix}\right] = \left[\begin{matrix}\frac{x}{\sqrt{x^{2} + y^{2}}}\\\frac{y}{\sqrt{x^{2} + y^{2}}}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{2}}{2}\\\frac{\sqrt{2}}{2}\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ \frac{3 \sqrt{10}}{10} \]
    D_u f = ∇f · u.Reviewed
Answer \( \frac{3 \sqrt{10}}{10} \)
A contour map of f(x, y) = sqrt(x² + y²): each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.245

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x^{2} y - y^{3}\right)\\\frac{\partial}{\partial y} \left(x^{2} y - y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}2 x y\\x^{2} - 3 y^{2}\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}-2\\-2\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{5}{13}\\\frac{12}{13}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ - \frac{34}{13} \]
    D_u f = ∇f · u.Reviewed
Answer \( - \frac{34}{13} \)
A contour map of f(x, y) = x²·y - y³: each curve joins points where f has the same value. At (-1, 1) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.279

medium Video
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
  1. \[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y^{2} + 2 x\right)\\\frac{\partial}{\partial y} \left(x y^{2} + 2 x\right)\end{matrix}\right] = \left[\begin{matrix}y^{2} + 2\\2 x y\end{matrix}\right] \]
    ∇f.✓ Proved
  2. \[ \left[\begin{matrix}18\\24\end{matrix}\right] \]
    ∇f at the point.Reviewed
  3. \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
    The unit direction u.Reviewed
  4. \[ 6 \sqrt{5} \]
    D_u f = ∇f · u.Reviewed
Answer \( 6 \sqrt{5} \)
A contour map of f(x, y) = x·y² + 2x: each curve joins points where f has the same value. At (3, 4) the gradient arrow points straight uphill, across the contours, and the direction of travel is drawn beside it.

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More gradient and directional derivatives practice

Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.10medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.13medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.15medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.16medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.18medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Problem 10.19medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.23medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.69medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.70medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.72medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.74medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.75medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.76medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.78medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.79medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.80medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.81medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.83medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.146medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.147medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Problem 10.149medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.150medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.151medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Problem 10.152medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.153medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.178medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.179medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.180medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.181medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.182medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.183medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.184medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Problem 10.185medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Problem 10.211medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.212medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.215medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.216medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.218medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.243medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.244medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.247medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.249medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.250medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.275medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Problem 10.276medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Problem 10.277medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.278medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Problem 10.280medium Video✓ Every equation proved
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Problem 10.282medium Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.22hard Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Problem 10.148hard Video✓ Nihil obstat
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Problem 10.248hard Video✓ Nihil obstat