Check my answer for directional derivative!!?

Question

Can someone check my answer for this question:
Find the directional derivative of f(x,y) = x/(x+y) at P(1,0) in the direction of Q (-1,-1).
Ok so the answer I am getting is 1/sqrt(2) but the answer at the back of my book is 1/sqrt(5). I have rechecked all my workings but still can't get the solution which matches the answer in the book. Can someone tell me if the answer I got is right...thanks :):):)

Answer 1

There are lots of ways of doing this problem, but since we are looking for possible mistakes, let's be as specific and direct (as opposed to abstract) as possible.

Suppose you are taking a step D in the direction of Q from the point P, then:

f(P+D) = f(1 - D/sqrt(2), -D/sqrt(2))

and the directional derivative =

(1/D)(f(P+D) - f(P))

When computing f(P+D) - f(P) you need to keep terms in D but no higher.

f(P+D) = (1 - D/sqrt(2)) / ((1 - D/sqrt(2)) - D/sqrt(2)) or

f(P+D) = (1 - e)/(1 - 2e) = 1 + e + 2e^2 + 4e^3 + ...

so f(P+D) - f(P) = (1 + e) - 1 = e

where e = D/sqrt(2)

so (1/D)(f(P + D) - f(P)) looks like 1/sqrt(2) to me too.

HTH.

Answer 2

The directional derivative interior the direction v is: df/dl = (?f/?x)(dx/dl) + (?f/?y)(dy/dl) (a million) the place dl is an infinitesimal length interior the direction of v=(-a million,2). this is obvious that a vector: v^ = (dx/dl)i + (dy/dl)j (2) is the *unit* vector interior the direction v (the place i and j are unit vectors in x and y guidelines). We basically word that: dx^2 + dy^2 = dl^2 (3) then sq. and upload the aspects of (2) to offer: (dx/dl)^2 + (dy/dl)^2 = (dx^2 + dy^2)/dl^2 = a million (4) So the unit vector (2) is: v^ = (dx/dl)i + (dy/dl)j = (-i + 2j)/?(a million^2 + 2^2) => v^ = -(a million/?5)i + (2/?5)j (5) because we'd write the gradient of f as: grad(f) = ?f = (?f/?x)i + (?f/?y)j (6) we'd write the directional derivative interior the direction v because of the fact the dot product (applying (a million), (2) and (6)): df/dl = v^ · ?f (7) So we'd desire to locate the gradient of f applying (6) then dot product the effect with v^ in (5), then we can plug interior the x and y values 2 and a million for the element (2,a million): ?f = (2x/(x^2 + y^2))i + (2y/(x^2 + y^2))j (8) Now dot this with v^ in (5): df/dl = (-(a million/?5)i + (2/?5)j)·( (2x/(x^2 + y^2))i + (2y/(x^2 + y^2))j ) (9) df/dl = (4y - 2x)/(x^2 + y^2)?5 Plug in x = a million and y = 2 and we get: *************** df/dl(2,a million) = (4(a million) - 2(2))/(2^2 + a million^2)?5 = 0 *************** you're ideal! this is 0, nicely carried out!