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

[Odyssey]

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!

 

[TD]

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?

 

[Odyssey]

So I guess the key is to find the gradient of f at the point?

 

[TD]

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?

 

[HallsofIvy]

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.