Calculus: Analytic Geometry and Calculus, with Vectors

3.5 Difference quotients and derivatives 161

9 Calculate

d 1 - x2 d

dx- (^1) + x2' dx(1 + x2)
by the quotient formula and by the product formula. Make the results agree.
10 It can be observed that the sum of the first two of the expressions
x2 1
2 + 1' x2 +
1, (^1) { x2
7isx
the third. Find the derivatives of these things and check the results by show-
ing that the sum of the first two is the third.
11 The formulas
d. d
dxsin x = cos x, dxcos x = - sin x
will be proved in Section 8.1. Copy them on a nice clean piece of paper, and
take a casual look at the calculations
d d sin x
tan x - =
cos2x+sin2x 1
cos2 x cos2 x
sin2 x - cos2 x - 1
sec2 x
dx ax cos x
d d cosx
dx-cot x-dx sin x sin2 x sin2 x - csc2 x
dx sec x = dx (cos x)'1 =-(cos x)-2(- sin x) =cos x
csin x
os x sec x tan x
d csc x = d (sin x)-1 = -(sin x)-2(cos x) =sinxsinsx =- csc x cot X.
Then, with the calculations out of sight, try to reproduce them.
12 Show that
dax+b ad-bc

dxcx+d

_

(cx -+d)*

13 Supposing that n is a positive integer and x 0 1, show how the identity

- 1

(1) 1 +x+x2+x3+ +xn =

xn+l

x- 1

can be used to obtain the less elementary identity

(2) 1 + 2x + 3x2 + .. + nxn'1 =

nxn+l - (n + 1)xn + I

(x-1)2

Multiply by x and differentiate again to obtain another identity.

14 Calculate the coordinates of the points on the graph of y = f(x) at which

f'(x) = 0 when

(a) f(x) = x3 - 3x As.: (-1,2) and (1,-2)

(b) f(x) = x3 - 3x + 2 .dns.: (-1,4) and (1,0)

(c) f (x) = 2x + 3 Ans.: None

(d) f(x) =1 + x2 .1ns.: (0,0)

(e) f(x) =

x and (1,1)

1 + x2

(J) f(x) = ax2 + bx + C Ans.: (-b/2a. -(b2 - 4,2r)/4,1)