MATH241 Lecture Notes - Lecture 14: Product Rule, Quotient Rule, Trigonometric Functions
9 May 2018
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MATH241 - Lecture 14 - Derivatives of Trigonometric Functions
3.3: Derivatives of Trigonometric Functions
Trigonometric Limits:
lim
θ → 0 θ
sinθ= 1
lim
θ → 0
θ
sinθ= 1
lim
θ → 0 θ
cosθ−1 = 0
Example
:
Find A) lim
x → 0 6x
sin6x
B) lim
x → 0 x
sin4x
C) lim
x → 0 3x
tanx
Solution:
A) Let , then if , , so xθ = 6 x→ 0 θ → 0 lim
x → 0 6x
sin6x= lim
θ → 0 θ
sinθ
= 1
B) lim
x → 0 x
sin4x= lim
x → 0 1
4·4x
sin4x
= 4 lim
x → 0 4x
sin4x
= 4 (1)= 4
C) lim
x → 0 3x
tanx = lim
x → 0 3x
( )
sinx
cosx
= lim
x → 0
sinx
3x cosx
= lim
x → 0 x
sinx ·1
3 cosx
= lim
x → 0 x
sinx lim
x → 0
1
3 cosx
1) = ( (1
3(1) )
=3
1
Derivatives of Sine
Let (x)inxf =s
(x)f′= lim
h → 0 h
sin(x + h) − sinx
in(x)inx cosh inh cosxs +h=s+s
= lim
h → 0 h
sinx cosh + sinh cosx − sinx
= lim
h → 0 h
sinxc osh − sinx + sinh cosx
= lim
h → 0 h
sinx cosh − sinx + lim
h → 0 h
sinh cosx
osx= lim
h → 0 h
sinx(cosh − 1) +clim
h → 0 h
sinh
inx osx=slim
h → 0 h
cosh−1 +clim
h → 0 h
sinh
osx= (sinx) (θ) + c(1)
(the derivative of sinx is cosx)osx=c
Example
:
Differentiate
A) (x)inxf =s+ex+x2
B) (x)sinxg =ex
C) (x)f=x2
sinx
Solution:
A) (x)osx xf′=c+ex+ 2
B) (x)sinx cosxg′=ex+ex
(sinx osx)= ex+c
C) (x)f′=(x)
22
(cosx)(x) − (sinx)(2x)
2
=x4
x cosx − 2x sinx
2
=x3
xcosx − 2sinx
Derivative of Cosine
Let (x)osxf =c
(x)f′= lim
h → 0 h
cos(x + h) − cosx
Document Summary
Math241 - lecture 14 - derivatives of trigonometric functions. 0 sin = 1 sin = 1 cos 1 = 0. C) lim sin6x x 0 6x sin4x x tanx x 0 3x. B) lim x 0 sin4x = lim x. C) lim x 0 3x tanx = lim sinx cosx x 0 3x sinx. = s inx sin(x + h) sinx h s in(x inx cosh. = c osx osx (1) (the derivative of sinx is cosx) = x2 (x) f g (x) f (x) + ex + 2 x cosx osx) (cosx)(x ) (sinx)(2x) 2 x cosx 2x sinx x4 xcosx 2sinx x3. = c f osx cos(x + h) cosx h f (x) = lim h 0 c os(x osx cosh. = (cosx) (0) (sinx) (1) (the derivative of cosx is -sinx) A) cosx) dx ( sinx (cosx)(cosx) (sinx)( sinx) cos x2. 2 cosx) dx ( 1 (0)(cosx) (1)( sinx) cos x2 d dx (secx) = d.
