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Dan Dolderman
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Mathematics

MAT237Y1

Dan Dolderman

Fall

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Calc 1
Mon 9/29/08
3.5 - Trig stu▯
All right, time for some derivatives of trig functions! Before we begin, I want to remind you of two
pretty imporant limits. They are limits that you should just remember because they like to creep
up every once in a while...like today! Here they are:
sin(x)
lim = 1
x!0 x
1 ▯ cos(x)
lim = 0
x!0 x
Both of these limits are going to be used to ▯nd the derivative of sin(x). One other thing we’re
going to need is the addition formula for sin(x), and that goes by:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
So, without further ado, let’s ▯gure this thing out.
d sin(x) = lim sin(x + h) ▯ sin(x)
dx h!0 h
sin(x)cos(h) + cos(x)sin(h) ▯ sin(x)
= lim
h!0 h
sin(x)cos(h) ▯ sin(x) cos(x)sin(h)
= lim +
h!0▯ h h ▯
cos(h) ▯ 1 sin(h)
= lim sin(x) + cos(x)
h!0 h h
= 0 + cos(x)
= cos(x)
Okay, so this means that the derivative of sin(x) is cos(x). I could go ahead and do the exact same
thing again for the derivative of cos(x), but it would just be a big fat waste of time, and I’d
basically be writing almost the exact same stu▯. So, I’ll just tell you that
d cos(x) = ▯sin(x)
dx
Notice that it’s negative sin(x).
Oh, that’s pretty cool! So, what if we keep going?
d2 d
sin(x) = cos(x) = ▯sin(x)
dx 2 dx
d3 d
sin(x) = ▯ sin(x) = ▯cos(x)
dx 3 dx
Can anybody guess what happens next?
d4 d
dx4sin(x) = dx ▯ cos(x) = ▯(▯sin(x)) = sin(x)
1 Whoa, that’s pretty neat! So that means that every forth derivative, sin(x) repeats itself! The
same thing happens with cos(x), too!
54
d
Example: ▯nd dx 54cos(x).
Solution: If you’d really like to, you can go ahead and take the derivative of cos(x) 54 times. Or,
you can just remember that cos(x) repeats itself every four derivatives, so let’s divide this guy by
4, and see what the remainder is! 54 = 13 ▯ 4 + 2. So, we can go ahead and skip taking the
derivative 52 times (13 ▯ 4), and just worry about that remainder of 2! In other words,
d54 d 2 d
cos(x) = cos(x) = ▯ sin(x) = ▯cos(x)
dx54 dx2 dx
Example: let f(x) = xcos(x). Find f (x).
Solution: Oh, snap! Product rule!
0
f (x) = cos(x) + x(▯sin(x)) = cos(x) ▯ xsin(x):
All right, now let’s go on to the other trig functions. Rather than doing all that limit junk again
(the de▯nition of the derivative), let’s just use the fact that each of the other trig functions are
based on sin(x) and cos(x)! Therefore, using the quotient rule,
d d sin(x)
dx tanx = dx cos(x)
cos(x)cos(x) ▯ sin(x)(▯sin(x))
= 2
cos (x)
cos (x) + sin2(x)
=
cos (x)
1
=
cos (x)
2
= sec (x)
Something like this may be on your exam. All i

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