Calculating the directional derivative of a function of two variables

[wshfulthinker]

Homework Statement

Consider the function:
z=f(x,y)= log(x^2 + y^2) (x,y)=/=(0,0)

Calculate the directional derivative of f(x,y) at (x,y)=(1,1) in the direction of the vector (1,2)


The attempt at a solution

When i tried to work out the unit vector from the point (1,1) to (1,2) i got (0,1).

I got partial derivative df/dx = 2x/(x^2 + y^2)
and partial derivative df/dy = 2y/(x^2 + y^2)

Then, for gradf at (1,1) i got (1,1)..

so, for directional derivative i got:

(gradf at (1,1)) x unit vector = (1,1).(0,1) = 1

But the answer is 3/root5

Does anyone know what i have done wrong? Thankyou

 

[tiny-tim]

Welcome to PF!

Hi wshfulthinker! Welcome to PF!

(have a curly d: ∂ and a square-root; √ and a grad: ∇ and try using the X² tag just above the Reply box )

Calculate the directional derivative of f(x,y) at (x,y)=(1,1) in the direction of the vector (1,2)
…
When i tried to work out the unit vector from the point (1,1) to (1,2) i got (0,1).

Nooo … you're msunderstanding "the direction of the vector (1,2)" …

it's not the point (1,2) (which is in direction (0,1) from (1,1), as you say) …

(1,2) is the actual direction that you're taking the derivative along.

(otherwise, your method is ok )

 

[wshfulthinker]

Hi, thanks for the welcome and showing me the symbols!

I don't really get it though! where do i use the point (1,2). I'm not even sure what i worked out, i followed the method that were in my lecture notes which were worded almost the exact same way as my actual question (except it said find instead of calculate - i don't know if that means it's different)

 

[tiny-tim]

I don't really get it though! where do i use the point (1,2).

grrr! it's not a point!

it's a direction … use it instead of your (0,1).

 

[wshfulthinker]

Okay okay... so i am pretty crap with vectors! But i got the answer finally! :D I found a book which wrote the direction in the i + j form which made more sense to me and didn't make me think it was a point. And yes, i got the answer so i think i kind of understand it now... Thankyou! :)