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Help in calculus problem!!

Hi, can anyone teach me how can I solve this problem? 
 
For the surface: 
g(x,y) = In(y + x2) + y
 
-- Find the directional derivative of ƒ(x,y) in the direction  u→ = i→ + 2j→    at the point where x = √e, y = e.
 
-- Find the direction u→ of maximum change at this point
 
-- Find the maximum change possible at this point. 
 
 
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Thank you so much and I will really appreciate your help. 
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1 Answer

The directional derivative is usually called the gradient.   The gradient operator (grad ) operates on a scalar function to produce a vector function.    In two dimensions we have
  
   grad( g(x,y)) =    d g(x,y) / dx  i  +  d g(x,y) / dy j      For g = ln(y + x2) + y   this works out to
 
   grad (g) =       [ 2x/(y + x2) ] i    + [ 1/(y + x2)  + 1 ] j    for x = √e   and y = e   this becomes
 
   grad (g )   = [ 1/√e ] i    +   [ 1/2e +1 ] j  
 
   The maximum change is the magnitude of this vector which is given by the usual formula for the
magnitude of a vector quantity i.e.   sqrt ( x_component2 + y_component2)   so
 
  maximum change =  sqrt ( 1/e + (1/2e +1)2 )
 
The direction of this max change is arctan( y_component / x_component)   so
 
 direction ( angle with respect to x axis ) =  arctan( ( 1/2e +1 ) /(1/√e) )

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