Find the directional derivative of f at the given point in the direction
indicated by the angle theta.
a) f(x, y) = (x^2 − y)^3, (3, 1), theta = 3pi/4.
b) f(x, y) = sin(x + 2y), (4,−2), theta = −2pi/3.
recall, the directional derivative of a function $\displaystyle f$ at the point $\displaystyle (x,y)$ in the direction of a unit vector $\displaystyle \bold{u} = \left< a,b \right>$ is given by
$\displaystyle D_u(x,y) = \nabla f \cdot \bold{u} = f_x(x,y)a + f_y(x,y)b$
now, our problem here is finding the unit vector. if you are completely lost, just draw a diagram with the unit circle to figure out the vector that points in the direction you want. recall, therefore, that unit vectors in the direction $\displaystyle \theta$ are given by $\displaystyle \left< \cos \theta , \sin \theta \right>$