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