If you have a dif/ble function of say two variables, its derivatives are given by the limits lim_{t->0}[f((x,y)+t(1,0))-f(x,y)]/t and lim_{t->0}[f((x,y)+t(0,1))-f(x,y)]/t --- that is, the rate of change of f in the directions ofi=(1,0) andj=(0,1) respectively. Notice that these limits are alike, but for the choice of vector along which you consider the change. Generalize this, by choosing the rate of change along any vectoruof unit length --- and the directional derivative is born!

What actually goes, is that the (familiar) derivatives are actually directional derivatives -- along the principal axes. By knowing these, you can find the differential of the function, df(x,y)=T, and then find the directional derivatives along any unit vectoruby calculating T(u). This is mostly what happens in applications.