what is a directional derivative/gradient?

Directional Derivative

The directional derivative del _(u)f(x_0,y_0,z_0) is the rate at which the function f(x,y,z) changes at a point (x_0,y_0,z_0)in the direction u. It is a vector form of the usual derivative, and can be defined as
del _(u)f=del f·(u)/(|u|)
(1)
=lim_(h->0)(f(x+hu^^)-f(x))/h,
(2)
where del  is called "nabla" or "del" and u^^ denotes a unit vector.
The directional derivative is also often written in the notation
d/(ds)=s^^·del
(3)
=s_xpartial/(partialx)+s_ypartial/(partialy)+s_zpartial/(partialz),
(4)
where s denotes a unit vector in any given direction and partialf/partialx=f_x denotes a partial derivative.
Let u^^=(u_x,u_y,u_z) be a unit vector in Cartesian coordinates, so
 |u^^|=sqrt(u_x^2+u_y^2+u_z^2)=1,
(5)
then


 del _(u^^)f=(partialf)/(partialx)u_x+(partialf)/(partialy)u_y+(partialf)/(partialz)u_z.
(6)

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