Chapter 10 Trigonometric functions - Ugrad.math.ubc.ca

Math 102 Notes Chapter 10
1
1
1
0.8
0.8
0.5
0.6
0.6
z
z
–3 –2 –1 0 1 2 3
x
0.4
0.4
–0.5
0.2
0.2
–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 –0.2 –0.1 0
0.1 0.2
–1
x
x
Figure 10.11: Zooming in on the graph of y = cos(x) at x = 0
f ′ f(x + h) − f(x)
(x) = lim
h→0 h
d sin(x) sin(x + h) − sin(x)
= lim
dx h→0 h
sin(x) cos(h) + sin(h) cos(x) − sin(x)
= lim
h→0
(
h
= lim sin(x) cos(h) − 1 + cos(x) sin(h) )
h→0 h
h
( ) ( )
cos(h) − 1
sin(h)
= sin(x) lim + cos(x) lim
h→0 h
h→0 h
= cos(x) (10.1)
Observe that the limits described in the preceding section were used in getting to our final result.
A similar calculation using the function cos(x) leads to the result
d cos(x)
dx
= − sin(x).
(The same two limits appear in this calculation as well.)
We can now calculate the derivative of the any of the other trigonometric functions using the
quotient rule. For example, let us consider the derivative of y = tan(x):
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