# Math Help - directional derivative???

1. ## directional derivative???

Hi,
Can you explain to me what is directional derivatives? I am so confused with this concept. Thank you very much.

2. 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 of i=(1,0) and j=(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 vector u of 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 vector u by calculating T(u). This is mostly what happens in applications.