Question

The derivative of f(x,y) at P(1,2) in the direction of i+j is 2sqrt(2) and in the direction of -2j is -3. What is the derivative of f in the direction of -i-2j?

Answer 1

Suppose the gradient of f is (x1,y1) at (1,2)
then
(x1,y1) . (1,1)/sqrt(2) = 2 sqrt(2) Hence
x1+y1 = 4
from the second directional derivative, we get
-2 y1 =-6
Hence the gradient of f at (1,2) is (1,3)

Now compute
(1,3) . (-1,-2) and divide sqrt(5) to get
\[-\frac{7}{\sqrt{5}}
\]

Answer 2

We used that the directional derivative in the direction of a vector u
is the dot product of the gradient of at the point and a unit vectr v in the direction of u.

Answer 3

thank you