Directional derivative

In summary, The directional derivative of f(x, y) = (xy)1/2 at P(6, 6) in the direction of Q(2, 9) is -2/5+3/10.
  • #1
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Homework Statement



Find the directional derivative of f(x, y) = (xy)1/2 at P(6, 6) in the direction of Q(2, 9)

Homework Equations



D_uf(x,y) = [tex]\nabla[/tex]f*u : dot product

The Attempt at a Solution



[tex]\nabla[/tex]f = <y1/2/(2x1/2),x1/2/(2y1/2)>

vector PQ = Q-P = <2-6,9-6> = <-4,3>

unit vector u of PQ is
u = PQ/|PQ| = <-4,3>/(4^2+3^2)1/2 = <-4/5 , 3/5>

D_uf(x,y) = <y1/2/(2x1/2),x1/2/(2y1/2)> * <-4/5 , 3/5> = -2y1/2/(5x1/2) + 3x1/2/(2y1/2)>


so D_uf(6,6) = -2/5+3/2

Somehow this isn't the right answer so where did I go wrong?

Thanks!
 
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  • #2
What happened to the 5 in the denominator of 3/5?
 
  • #3
Dick said:
What happened to the 5 in the denominator of 3/5?

Hmm... that's a good question :P

D_uf(6,6) = -2/5+3/10
 

1. What is a directional derivative?

A directional derivative is a measure of how a function changes in a particular direction. It is the rate of change of a function at a given point in the direction of a specified vector.

2. How is a directional derivative calculated?

The directional derivative is calculated using the dot product of the gradient of the function and the unit vector in the desired direction. The formula is given as Df(x,y) = ∇f(x,y) · u, where ∇f(x,y) is the gradient vector and u is the unit vector in the direction of interest.

3. What is the significance of directional derivatives in mathematics?

Directional derivatives play a crucial role in optimization problems, such as finding the steepest descent or ascent of a function. They also have applications in physics, engineering, and other sciences where studying the rate of change in a particular direction is important.

4. Can the directional derivative be negative?

Yes, the directional derivative can be negative. It indicates that the function is decreasing in the direction of interest at the given point. A positive directional derivative indicates an increase in the function.

5. How is the direction of steepest ascent or descent determined using directional derivatives?

The direction of steepest ascent or descent is determined by finding the unit vector in the direction of the gradient vector. This unit vector will give the direction in which the function changes the most rapidly at a specific point.

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