Maths~Gradient

f(x,y) =xe^(2x+3y)

differentiate f with respect to x is f_x = e^(2x+3y) + 2xe^(2x+3y)
differentiate f with respect to y is f_y = 3xe^(2x+3y)

so, grad f = (f_x , f_y) = (e^(2x+3y) + 2xe^(2x+3y), 3xe^(2x+3y) )

grad f (1/2 , 1/3) = (2e^2, 1.5e^2)

When u is in the same direction of grad f, it has greatest rates of change
When u is in the opposite direction of grad f, it has least rates of change
When u is orthogonal to grad f, its directional derivative is zero

So, grestest rate of change is the same direction of grad f,
that is vector u = grad f (1/2 , 1/3) = (2e^2, 1.5e^2)
So, the greatest value is || u || = || grad f || = 2.5e^2