320 CHAPTER 17. THE DERIVATIVE OF A FUNCTION OF MANY VARIABLES

Proof: From the assumption that Df (g (x)) exists,

f (g (x+v)) = f (g (x))+Df (g (x))(g (x+v)−g (x))+o(g (x+v)−g (x))

= f (g (x))+Df (g (x))(Dg (x)v+o(v))+o(g (x+v)−g (x))

which by Lemma 17.7.7 equals

= f (g (x))+Df (g (x))Dg (x)v+Df (g (x))o(v)+o(v)

= f (g (x))+Df (g (x))Dg (x)v+o(v)

and this showsD(f ◦g)(x) = Df (g (x))Dg (x)

from the definition of the derivative and its uniqueness established in Theorem 17.3.3 onPage 301. ■

17.8 Exercises1. Suppose f : U → Rq and let x ∈U and v be a unit vector. Show that Dvf (x) =

Df (x)v. Recall that

Dvf (x)≡ limt→0

f (x+ tv)−f (x)

t.

2. Let f (x,y) =

{xysin

( 1x

)if x ̸= 0

0 if x = 0. Find where f is differentiable and compute the

derivative at all these points.

3. Let

f (x,y) =

{x if |y|> |x|−x if |y| ≤ |x|

.

Show that f is continuous at (0,0) and that the partial derivatives exist at (0,0) butthe function is not differentiable at (0,0).

4. Let

f (x,y,z) =

(x2 siny+ z3

sin(x+ y)+ z3 cosx

).

Find Df (1,2,3).

5. Let

f (x,y,z) =

(x tany+ z3

cos(x+ y)+ z3 cosx

).

Find Df (x,y,z).