# Directional derivative question

## Homework Statement

rate of change of $f(x,y) = \frac{x}{(1+y)}$ in the direction (i-j) at the point (0,0)

## The Attempt at a Solution

$∇f(x,y) = \frac{1}{(y+1)}\hat{i} - \frac{x}{(y+1)^2}\hat{j}$

$D_u = ( f_x, f_y) \bullet ( 1, -1 )$

$D_u = \frac{(y+x+1)}{(y+1)^2}$

Wolfram and the answer sheet is telling me that there should be a $\sqrt{2}$ in the denominator, but I can't figure out where it comes from, thanks.

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Dick
Science Advisor
Homework Helper

## Homework Statement

rate of change of f(x,y) = x/(1+y) in the direction (i-j) at the point (0,0)

## The Attempt at a Solution

grad(f(x,y)) = 1/(y+1)i - x/(y+1)^2j

Du = ( f_x, f_y ) dot ( 1, -1 )

Du = (y+x+1)/(y+1)^2

Wolfram and the answer sheet is telling me that there should be a sqrt(2) in the denominator, but I can't figure out where it comes from, thanks.

The vector you want to dot the grad with should be a unit vector pointed in the direction i-j. That's what 'in the direction' means.

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