Question

So, according to my book, directional derivatives and gradients are related as follows: If f is differentiable at (a,b) and \[u=u_1i+u_2j\] is a unit vector, then \[fu(a,b)=f_x(a,b)u_1+f_y(a,b)u_2 = gradf(a,b)*u\]

Then given the information below, can I compute the gradient as asked?

Answer 1

Please note that the u's, i's, and j's should have harpoons over them for vectors

Answer 2

I know from my book that \[grad f(a,b)=f_x(a,b)i+f_y(a,b)j\] so, I need to use what Im given to identify the directional derivatives at (5,7).

Answer 3

I am trying to find a link in my book, but does anyone have any idea if (and what it is) there is a link between directional derivatives and unit vectors? Im struggling to find this connection to backtrack from the unit vector or the given vector.

Answer 4

Anyone familiar with calculus that might know something to help?

Answer 5

Ok how about this given what we know, I am plugging in (5,7) for (a,b), and what I am given for u in where there are u's.. does seem ok to do?\[f_u(5,7) = \sqrt2 = f_x(5,7)u_1+f_y(5,7)u_2 = gradf(5,7)*u\]