Directional Derivatives

The idea will first be illustrated by showing an example:

The directional derivative of $f(x,y)=x^2y^4$ at $p=(p_1, p_2)$ in direction $u=(u_1, u_2)$ is:

$\nabla_{(u_1, u_2)}f = 2 (p_1)(u_1) + 4 (p_2)^3(u_2)$