22.3. EXERCISES 467

7. Use the definition of the derivative to find the 1×1 matrix which is the derivative ofthe following functions.

(a) f (t) = t2 + t.(b) f (t) = t3.(c) f (t) = t sin(t).(d) f (t) = ln

(t2 +1

).

(e) f (t) = t |t|.

8. Show that if f is a real valued function defined on (a,b) and it achieves a localmaximum at x ∈ (a,b), then D f (x) = 0.

9. Use the above definition of the derivative to prove the product rule for functions of 1variable.

10. Let f (x,y) = xsin(y). Compute the derivative directly from the definition.

11. Let f (x,y) = x2 sin(y). Compute the derivative directly from the definition.

12. Let f (x,y) =(

x2 + yy2

). Compute the derivative directly from the definition.

13. Let f (x,y) =(

x2yx+ y2

). Compute the derivative directly from the definition.

14. Let f (x,y) = xα yβ . Show D f (x,y) =(

αxα−1yβ xα βyβ−1).

15. Let f (x,y) =(

x2 sin(y)x2 + y

). Find Df (x,y).

16. Let f (x,y) =√

x 3√

y. Find the approximate change in f when (x,y) goes from (4,8)to (4.01,7.99).

17. Suppose f is differentiable and g is also differentiable, g having values in R3 and fhaving values in R. Find D( fg) directly from the definition. Assume both functionsare defined on an open subset of Rn.

18. Show, using the above definition, that if f is differentiable, then so is t → f (t)n forany positive integer and in fact the derivative of this function is n f (t)n−1 f ′ (t).

19. Suppose f is a scalar valued function of two variables which is differentiable. Showthat (x,y)→ ( f (x,y))n is also differentiable and its derivative equals

n f (x,y)n−1 D f (x,y)

20. Let f (x,y) be defined on R2 as follows. f(x,x2

)= 1 if x ̸= 0. Define f (0,0) = 0,

and f (x,y) = 0 if y ̸= x2. Show that f is not continuous at (0,0) but that

limh→0

f (ha,hb)− f (0,0)h

= 0

for (a,b) an arbitrary unit vector. Thus the directional derivative exists at (0,0) inevery direction, but f is not even continuous there.