Calculus: Analytic Geometry and Calculus, with Vectors

8.1 Trigonometric functions and their derivatives

1.0

0.8

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8

0.7

0.6

0.6

0.4

0.5

0.4

0.3

02

0.1

0 0.2 0.4 0.6 0.8 1.0

Figure 8.17

443

Figure 8.171

Figure 8.171. For each integer n, the interval Ix- n71 < 7/2 contains

an exact copy of the graph in the interval jxI < 7r/2. The lines

x=n7r±27r

are all vertical asymptotes of the graph, and tan (nlr ± a/2) is undefined.

We should now be quite familiar with the fact that each formula for

a derivative has a chain extension. The following list contains the chain

extensions of formulas for derivatives of the six trigonometric functions

and three additional formulas. All of these must be learned.

dx sin u = cos u

dx

du d

dx

cot u

d du d

dx

cos u = - sin u

dx dx

sec u

z,tan u = sect udx

du d

dx

csc u

d

TX

u'' = nun-1

dx' dx

e°

=

eu

dx'

Problems 8.19

• csc2

du

u

dx

du

sec u tan n

TX

du

• csc u cot u
  dx
  1 du
  d
  YX-

log u =

u dx

1 With all graphs and tables out of sight, make the pretense that it has been

forgotten whether the derivative with respect to x of sin x is sin x or - sin x or

cos x or - cos x. Sketch graphs of sin x and cos x and make these graphs give

the correct answer.

2 Explain how you should modify the procedure for sketching the graph of