Question 1106994
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<pre>
If "s" is the parameter on the straight line x = y along the vector (1,1), then we have  


    s = {{{x*sqrt(2)}}}, y = {{{s*sqrt(2)}}},   or, equivalently,  x = {{{s/sqrt(2)}}},  y = {{{s/sqrt(2)}}}.


Therefore, the numerator is {{{x^2}}} + {{{2y^2}}} = {{{(s/sqrt(2))^2}}} + {{{2*(s/sqrt(2))^2}}} = {{{s^2/2}}} + {{{(2s^2)/2)}}} = {{{(3s^2)/2}}},  


while the denominator is x + y = {{{s/sqrt(2)}}} + {{{s/sqrt(2)}}} = {{{(2s)/sqrt(2)}}}.


Then the ratio itself is

f(x,y) = f(s) = {{{(((3*s^2)/2))/(((2*s)/sqrt(2)))}}} = {{{(3*sqrt(2)*s)/4}}}.


Thus the function f(s) is LINEAR on s along this direction, and is zero at  x= y= 0= s by the definition, which is consistent with the linear behavior.


So (and therefore), the derivative  {{{(df)/(ds)}}}  DOES EXIST  and is equal to {{{(3*sqrt(2))/4}}}.
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To avoid misunderstanding, let me note (highlight/underline) that for the given function the derivative "along a direction" 
DEPENDS on direction, so the function f(x,y) is NOT differentiate at (0,0) in the classic sense as a function of two variables.


It is ONLY differentiate "along a direction", and is a classic example of such a function.