4. [QUOTIENT RULE] Let f : Rn -> R be differentiable at a with f

(a) f 0.

(a) Show that for Ilhll sufficiently small, f(a+h) f 0.

(b) Prove that D f(a)(h)/llhll is bounded for all h ERn \ {o}.

(c) 1fT:= -Df(a)IP(a), show that 1 __ 1 __ T(h) = f

(a) - f(a + h) + D f(a)(h)

f(a+h) f

(a) f(a)f(a+h)

(f(a+h) - f(a)) Df

(a) (h)

+ j2(a)f(a+h)

for Ilhll sufficiently small.

(

d) Prove that 1 I f (x) is differentiable at x = a and

(1) Df(a)

D 7

(a) = - j2

(a) .

(e) Prove that if f and 9 are real-valued vector functions that are differentiable at some

a, and if g

(a) f 0, then D (L)

(a) = g(a)D f

(a) - f(a)Dg

(a) .

9 g2(a)