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YOU WILL LEARN ABOUT:

The directional derivative, which is a rate of change of a multivariable function in any direction.
Partial derivatives turn out to be directional derivatives along the coordinate axes.

The derivative of f(x,y) at the point (x,y) in the direction of the unit vector is:
  • Du→f(x,y)=limh→0f(x+ah, y+bh)−f(x, y)h

The rate of change of a measurement such as air pressure, rock density, and altitude depends on the path travelled, thus requiring a directional derivative for its computation.

As a rover on Mars travels in a particular direction, how do the atmospheric and geogaphic conditions change? To compute the answer, NASA scientists compute a directional derivative.
Geographic analysts measure changes in density in the ground with a directional derivative. This helps to find the best path to drill for oil.
Directional derivatives are used in the optimization calculations made by rescue robots, which are designed to find people trapped in a collapsed structure using the quickest and safest path possible.

Click on the tabs on the right to watch videos on rotating a region about the x-axis and y-axis.

DIRECTIONAL DERIVATIVES
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Rotate view 360o
Display direction plane

(0.5, 1.75)

(1, -1)

(0, 0)
fx,y=x+xy+y
fx,y=x2yx4+y2if x,y≠0,00if x,y≠0,0
fx,y=xyx2+y2 if x,y≠0,00if x,y=(0,0)
South
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u→=d→d→=
〈 0.706,0.708 〉
d→=
〈 , 〉
Select one of the three points, then specify the direction of the unit vector by dragging the yellow arrow or typing in a vector.
1 2 3
1. Moving northwest from the origin, does the function value increase or decrease?
2. Moving from the point (1, -1), in which direction is the function increasing?
3. When moving from the point (0.25, 1), in which direction is the function decreasing? Check all that apply.
Select one of the three points, then specify the direction of the unit vector by dragging the yellow arrow or typing in a vector.
1 2 3
1. At the point (1.5, 0), in which direction is f (x,y) increasing? Check all that apply.
2. At the point (0.5, 1), in which direction is f (x,y) decreasing? Check all that apply.
3. At the point (0,0), can we use the theorem that says Duƒ (0, 0) =
ƒx (0, 0) a + ƒy (0, 0) b for a unit vector ? (Hint: Compare the directional derivative value in various directions with the partial derivative values at (0, 0).)
Select one of the three points, then specify the direction of the unit vector by dragging the yellow arrow or typing in a vector.
1 2 3
1. When moving through the point (-1, 0.75), in which of the following directions does the function decrease? Check all that apply.
2. At the point (0.01, 0.05), in which of the following directions does the function increase faster than the others?
3. At the point (0, 0), the directional derivative only exists along the x-axis and the y-axis (that is, in the north, south, east, and west directions). In which direction is the directional derivative 0? Select all that apply.
Result