Directional derivative and angles.

Consider a surface S given by the equation z = f(x,y) where f(x,y)= xe^(4x^2-y^2).

(a) Find the directional derivative Duf in the direction of u parallel to v = 3i+4j at the point A=(1,2).

(b) Find the angles THETAx and THETAy of the directional vector u in (x,y) plane relative to x and y axes, respectively, at which the maxim of Duf at the point A is reached.

For (a) I used the formula Duf(x,y) = Fx(x,y)cosTHETA + Fy(x,y)sinTHETA. I took the first order partial derivatives and plugged in the point A=(1,2). For THETA (and this is where I might have made a mistake) I used 3/4 because the directional derivative is parallel to v = 3i+4j.

theta is arctan (4/3)

To verify we can compute Df 2 ways using the formula you gave
and the fact cos*arctan(4/3) = 3/5 and sin(arctan(4/3) =4/5

Or we could use gradf*u where u is a unit vector since |3 i +4j| =5

we have u = 3/5 i +4/5 j as before

by the way the directional derivative is a number so it can't be parallel to u