Directional derivative

In summary, the conversation discussed finding the direction and values of $D_{u}f$ at a given point, using the formula for directional derivatives and the concept of gradients. It was mentioned that the maximum and minimum values of $D_{u}f$ occur in the direction of the gradient, while the direction of the minimum value is the opposite direction. The process of finding these values was also briefly explained.
  • #1
shorty1
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Let f(x,y) = (x-y)/(x+y). find the directions u and the values of $D_{u}f $ (-1/2 , 3/2) for which $D_{u}f $ (-1/2 , 3/2) is largest, and is smallest.

How do i go about that? I did it for when $D_{u}f $ (-1/2 , 3/2) = 1 and got $D_{u}f $ (-1/2 , 3/2) = 1 and got u=j and -i. This was after i equated partials for x and y at the point (-1/2, 3/2) and then substituted that into the formula for directional derivative, letting u = $u_{1}i + u_{2}j$i assume i can do the same to calculate when $D_{u}f $ (-1/2 , 3/2) = 0 and -2, can i? but i am not sure how to work it for when it is largest and smallest.

Any help will be appreciated.
 
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  • #2
For a function of two variables, f(x,y), its gradient is [tex]\nabla f= f_x\vec{i}+ f_y\vec{j}[/tex]. It is also true that a unit vector in a direction that makes angle [tex]\theta[/tex] with the x-axis is [tex]cos(\theta)\vec{i}+ sin(\theta)\vec{j}[/tex]. The directional derivative of f, in the direction of the unit vector [tex]\vec{u}[/tex] is [tex]D_{\vec{u}}f= \nabla f\cdot\vec{u}= f_xcos(\theta)+ f_ysin(\theta)[/tex].

Now, find the direction in which that is maximum or minimum by taking the derivative with respect to $\theta$ and setting it equal to 0:
[tex]-f_xsin(\theta)+ f_ycos(\theta)= 0[/tex].
That is the same as [tex]\frac{f_y}{f_x}= \frac{sin(\theta)}{cos(\theta)}= tan(\theta)[/tex]. That is, the max and min lie in the direction [tex]\theta[/tex] such that [tex]tan(\theta)= \frac{f_y}{f_x}[/tex]. But, looking at the right triangle having legs [tex]f_x[/tex] and [tex]f_y[/tex] and so hypotenuse [tex]\sqrt{f_x^2+ f_y^2}[/tex]. That is, the direction in which the derivative is largest is precisely the direction in which the gradient is pointing. And the direction in which the derivative is least is just the opposite direction. (Of course, [tex]D_u f= \nabla f\cdot \vec{u}= 0[/tex] just says the vector [tex]\vec{v}[/tex] is perpendicular to the gradient.
 
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FAQ: Directional derivative

What is a directional derivative?

A directional derivative is a measure of the rate of change of a function in a specific direction, defined as the slope of the tangent line to the function in that direction.

How is a directional derivative calculated?

The directional derivative is calculated using the dot product of the gradient vector of the function and a unit vector representing the desired direction.

What is the significance of directional derivatives?

Directional derivatives are useful in understanding how a function changes in a particular direction, which can be applied in various fields such as physics, engineering, and economics.

What is the difference between a partial derivative and a directional derivative?

A partial derivative measures the rate of change of a function with respect to one variable, while a directional derivative measures the rate of change in a specific direction.

How can directional derivatives be used in optimization problems?

In optimization problems, directional derivatives can be used to find the direction of steepest ascent or descent, which can help in determining the maximum or minimum values of a function.

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