Finding the Derivative of f at (1,2) in a Different Direction

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In summary, the conversation discusses finding the directional derivative of a function at a given point in different directions. The derivative is given in two directions, and the third direction can be found by using a linear combination of the first two directions. The conversation also mentions finding the gradient of the function at the given point and using that to find the derivative in any direction. The process may require some thought and understanding of how a directional derivative relates to the gradient.
  • #1
Odyssey
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Greetings, I got a Q on directional derivatives.
The derivative of f(x,y) at (1,2) in the direction of [1,1] is 2(2)^.5 and in the direction of [0, -2] is -3. What is the derivative of f in the direction of [-1, -2]??

Thank you!
 
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  • #2
Calculate the vectors which give the direction from your point (1,2) to the three given points. Can you then find a relation between the first two and the last directional vectors and use that result?
 
  • #3
So I guess the key is to find the gradient of f at the point?
 
  • #4
Yes, but you don't have any information about the function so explicitly finding the gradient won't be easy. You know the answers in two directions though, can't you form your third direction as a lineair combination of the first two and then use that relation on the results?
 
  • #5
You are given the derivative in the direction of (0, -3) so you can immediately get the partial derivative with respect to y (that is the derivative in the direction of (0, 1). It will take a bit of thought to use the derivative in the direction (1, 1) along with that to get the partial derivative with respect to x but it can be done (remember how a directional derivative relates to the gradient). After you get that, you can find the gradient and then get the derivative in any direction.
 

1. What is a directional derivative?

A directional derivative is a measure of how a function changes in a particular direction, given a starting point. It is typically used in multivariate calculus to calculate the rate of change of a function along a specific direction or vector.

2. How is a directional derivative calculated?

A directional derivative is calculated by taking the dot product of the gradient of the function and the unit vector in the specified direction. This represents the rate of change of the function in that direction at the given point.

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

A directional derivative is a generalization of a partial derivative, which only considers the change of a function in one specific direction. A partial derivative, on the other hand, considers the change of a function with respect to one of its variables while holding others constant.

4. Can a directional derivative be negative?

Yes, a directional derivative can be negative. This indicates that the function is decreasing in the specified direction at the given point.

5. In what fields is the concept of directional derivatives commonly used?

Directional derivatives are commonly used in fields such as physics, engineering, and economics. They are also used in computer graphics and image processing to calculate the rate of change of a function along different directions.

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