The Directional Derivative   

If f(x,y) is differentiable at a point p = ( p,q) and if u = á m,n ñ is a unit vector, then the derivative of f at p in the direction of u is defined to be
(Duf) (p) =
lim
h®0+ 
    f( p+hu) - f( p)
h
  
(1)
We say that Duf is the directional derivative of f in the direction of u.   For example, in the direction of i we have
(Di f) (p) =
lim
h®0+ 
    f( p+hi) - f( p)
h
  = 
lim
h®0+ 
    f(p+h,q) - f(p,q)
h
 = fx(p,q)
Indeed, the directional derivatives in the directions of i and j, respectively, are the first partial derivatives
Di f = f
x
         and     Dj f =  f
y
  

The directional derivative can be interpreted geometrically via vertical slices of the surface z = f(x,y), where a vertical slice is a curve formed by the intersection of the surface z = f(x,y) with the vertical plane through a line r(t) = p+ut in the xy-plane.
   
Specifically, the z-coordinate of the vertical slice is z(t) = f(p+tu), which in non-vector form is z(t) = f(p+mt, q+nt).  The definition of the directional derivative is equivalent to 
  (Du f) (p)  =  z' (0)  

Since u is a unit vector, the point r(h) is a distance h from r(0) . Thus, a "run" of h causes a "rise" of z(h) - z(0).

Thus, as h approaches 0, the slope of the secant line (in blue) approaches the slope of the tangent line (in red). That is, the slope of the tangent line at r(0) = p = (p,q) is 
 z' (0)  = 
lim
h® 0 
  z(h) - z(0)
h
where z = f(p+mt, q+nt).  Moreover, since r(t) = p+ut implies that r' (0) = u and since (Du f) (p)  =  z' ( 0), the chain rule implies that
(Du f) (p)  =    dz
dt
 ê
ê

t = 0 
 = Ñf · r' (0) = Ñf · u
That is, the slope of the tangent line is Ñf · u, where u is a unit vector in the direction of r(t).

Theorem 7.1    The directional derivative of f(x,y) in the direction of a unit vector u is given by
Duf = Ñf · u
Moreover, Duf is the slope of the tangent line to the curve formed by the intersection of z = f( x,y) and the vertical plane through a point p parallel to a unit vector u in the xy-plane.


Drag red point to change (p,q). Notice that direction of derivative stays fixed.

Moreover, theorem 7.1 confirms what we alluded to in section 3, which is so that fx yields slopes of vertical slices parallel to the xz-plane and fy yields slopes of vertical slices parallel to the yz-plane.       

EXAMPLE 3    Find the derivative of f( x,y) = 1.1x2 - 0.1xy in the direction of v = á 3,4 ñ.       

Solution: Since v is not a unit vector, we first finds its direction vector:
u =  1
v
 v =  1
5
 á 3,4 ñ =
 3
5
 ,  4
5
   = á 0.6, 0.8 ñ  
The gradient of f is Ñf = á 2.2x - 0.1y, -0.1x ñ, so that
Du  f   =   á 2.2x -0.1y, -0.1x ñ · á 0.6, 0.8 ñ
=   0.6( 2.2x - 0.1y) + 0.8 (-0.1x )
=   1.24x - 0.06y
The applet below shows Du  f  at the point (1,1).

LiveGraphics3d Applet
Drag red point to change (p,q). Notice that direction of derivative stays fixed.

  Check your Reading: What is Du f at the point ( 1,1) ?