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c Dr Oksana Shatalov, Fall 2013
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Section 3.4: Derivatives of Trigonometric Functions
It is important to remember that everything for six trigonometric functions (sin x, cos x, tan x, cot x, csc x, sec x)
will be done in radians.
EXAMPLE 1. Compute:
(a) lim sin x =
(b) lim cos x =
x→0
THEOREM 2.
lim
x→0
x→0
sin x
= 1,
x
lim
x→0
cos x − 1
= 0.
x
Proof
EXAMPLE 3. Find these limits:
sin(5x)
x→0
x
(a) lim
sin(9x)
x→0 sin(7x)
(b) lim
x
x→0 sin(4x)
(c) lim
Conclusion: If a, b 6= 0 then
lim
x→0
sin(ax)
=
x
,
lim
x
x→0 sin(ax)
=
,
lim
sin(ax)
x→0 sin(bx)
=
c Dr Oksana Shatalov, Fall 2013
(d) lim
2
1
x→0 x2 cot2 (3x)
cos x − 1
x→0 sin x
(e) lim
EXAMPLE 4. Find the following derivatives:
(a)
d
sinx =
dx
Remark Similarly one can get (cos x)0 = − sin x.
(b)
d
tanx =
dx
Derivatives of Trig Functions (memorize these!)
d
sinx =
dx
d
cosx = − sin x
dx
d
tanx =
dx
d
cscx = − csc x cot x
dx
d
secx = sec x tan x
dx
d
cotx = − csc2 x
dx
c Dr Oksana Shatalov, Fall 2013
EXAMPLE 5. Find the derivative of these functions.
√
(a) y = cot x + 5 sec x + x x
(b) f (x) =
cos x
1 + sin x
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