1549901369-Elements_of_Real_Analysis__Denlinger_

6.1 The Derivative and Differentiability 303

(b) If 3 o > 0 :1 f is differentiable on (xo, xo+o) and continuous from the right
at xo, and lim f'(x) exists, th en f~(xo) exists and equals lim f'(x).
x--+xci x--+xci

Proof. P ostponed to Section 6.4, Exercise 29. •

EXERCISE SET 6.1

1. Use Definition 6.1.1 to calculate the derivative of the given function fat
   X = X o.
   (a) f(x) = 3x^2 - 2x

1

(c) f(x ) = - (x =/= 0)

x

(e) f(x ) = v'2x + 3 (x > -~)

(b) f(x) = x^3

(d) f(x) = x +

1

(x =/= 1)

x-1

1

(f) f(x) = Vx (x > 0)

2. Use (alternate) Definition 6.1.6 to calculate the derivative of the given
   function f at x = x o.
   (a) f(x) = 4x^2 + 3x - 5 (b) f(x) = x^3

5

(c) f(x) = - (x =/= 0)

x

1

(d) J(x) = 5x + 4 (x =/= - t )

(e) f(x) = y'4x - 1 (x > ~)

1

(f) f(x) = ijX (x =/= 0)

3. Let f(x) = { x2 ~f x ~ 0,
   0 if x < 0
   (a) Sketch the graph off.
   (b) Prove that f is differentiable at 0.
   ( c) Sketch the graph of f'. Is f' continuous at O? (Prove or disprove.)
   ( d) Is f' differentiable at O? (Prove or disprove.)


4. Let f ( x) = { x

2

~f x ~ O, }. Prove that f is not differentiable at 0.

x ifx<O

5. Let f(x) = {.... '. Prove that f is differentiable at 0. Is
   x^2 if x is rational }...
   0 if x is irrat10nal
   f differentiable anywhere else? Explain.