450 CHAPTER 21. FUNCTIONS OF MANY VARIABLES

7. ∗Suppose f is a function defined on a set D and that a ∈ D is not a limit pointof D. Show that if I define the notion of limit in the same way as above, thenlimx→a f (x) = 5. Show that it is also the case that limx→a f (x) = 7. In otherwords, the concept of limit is totally meaningless. This is why the insistence that thepoint a be a limit point of D.

8. ∗Show that the definition of continuity at a ∈ D(f) is not dependent on a being alimit point of D(f). The concept of limit and the concept of continuity are related atthose points a which are limit points of the domain.

21.4 Directional and Partial Derivatives21.4.1 The Directional DerivativeThe directional derivative is just what its name suggests. It is the derivative of a function ina particular direction. The following picture illustrates the situation in the case of a functionof two variables.

vIn this picture, v ≡ (v1,v2) is a unit vector shown in the xy plane and x0 ≡ (x0,y0) is a

point in the xy plane with (x0,y0, f (x0,y0)) being the point on the surface where there is atangent line. When (x,y) moves in the direction of v, this results in a change in z = f (x,y).The directional derivative in this direction is the slope of the tangent line shown in thepicture defined as

limt→0

f (x0 + tv1,y0 + tv2)− f (x0,y0)

t.

It tells how fast z is changing in this direction. A simple example of this is a person climb-ing a mountain. He could go various directions, some steeper than others. The directionalderivative is just a measure of the steepness in a given direction. This motivates the follow-ing general definition of the directional derivative when it is not possible to draw pictures.

Definition 21.4.1 Let f : U → R where U is an open set in Rn and let v be a unitvector. For x ∈ U, define the directional derivative of f in the direction v, at the point xas

Dv f (x)≡ limt→0

f (x+ tv)− f (x)t

.

Example 21.4.2 Find the directional derivative of the function f (x,y) = x2y in the direc-tion of i+j at the point (1,2).

First you need a unit vector which has the same direction as the given vector. Thisunit vector is v ≡

(1√2, 1√

2

). Then to find the directional derivative from the definition,

write the difference quotient described above. Thus f (x+ tv) =(

1+ t√2

)2(2+ t√

2

)and