Download Lecture 20 Slides - Department of Mathematical Sciences

MATH 12002 - CALCULUS I
§2.3 & §2.4: Derivatives of Trigonometric Functions
Professor Donald L. White
Department of Mathematical Sciences
Kent State University
D.L. White (Kent State University)
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Derivatives of Sine & Cosine
Our first two differentiation formulas for the trigonometric functions are
d
d
sin x = cos x and
cos x = − sin x.
dx
dx
The first of these is proved in the text; we will prove the second using the
definition of derivative. We will also need the angle sum formula for cosine,
cos(A + B) = cos A cos B − sin A sin B,
and the limits from Equation 6 and Example 11 of §1.4,
sin θ
cos θ − 1
= 1 and lim
= 0.
θ→0 θ
θ→0
θ
lim
D.L. White (Kent State University)
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Derivatives of Sine & Cosine
By definition of the derivative, we have
d
cos x
dx
=
=
=
=
=
=
=
D.L. White (Kent State University)
cos(x + h) − cos x
h
[cos x cos h − sin x sin h] − cos x
lim
h→0
h
cos x(cos h − 1) − sin x sin h
lim
h→0
h
cos h − 1
sin h
lim cos x
− sin x
h→0
h
h
cos h − 1
sin h
lim cos x
− lim sin x
h→0
h→0
h
h
cos h − 1
sin h
(cos x) lim
− (sin x) lim
h→0
h→0
h
h
(cos x)(0) − (sin x)(1) = − sin x.
lim
h→0
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Derivatives of Other Trigonometric Functions
Recall that the other trigonometric functions can be written in terms of
sin x and cos x:
tan x =
sin x
,
cos x
cot x =
cos x
,
sin x
sec x =
1
,
cos x
csc x =
1
.
sin x
We can use these relations and the derivatives of sin x and cos x to find
the derivatives of all of the trigonometric functions.
D.L. White (Kent State University)
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Derivatives of Other Trigonometric Functions
Using the Quotient Rule, we have
d
d sin x
tan x =
dx
dx cos x
d
d
dx (sin x) cos x − sin x dx (cos x)
=
cos2 x
Hence
d
dx
=
(cos x) cos x − sin x(− sin x)
cos2 x
=
cos2 x + sin2 x
1
=
= sec2 x.
2
cos x
cos2 x
tan x = sec2 x. Similarly,
D.L. White (Kent State University)
d
dx
cot x = − csc2 x.
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Derivatives of Other Trigonometric Functions
Again using the Quotient Rule, we have
d
1
d
sec x =
dx
dx cos x
d
d
dx (1) cos x − 1 · dx (cos x)
=
cos2 x
Hence
d
dx
=
(0) cos x − 1(− sin x)
cos2 x
=
sin x
1
sin x
=
·
= sec x tan x.
2
cos x
cos x cos x
sec x = sec x tan x. Similarly,
D.L. White (Kent State University)
d
dx
csc x = − csc x cot x.
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Derivatives of Other Trigonometric Functions
We now have
Derivatives of Trigonometric Functions
d
dx
sin x = cos x
d
dx
cos x = − sin x
d
dx
tan x = sec2 x
d
dx
cot x = − csc2 x
d
dx
sec x = sec x tan x
d
dx
csc x = − csc x cot x
Notes:
Derivatives involving cot x and csc x may show up in homework
problems on WebAssign occasionally due to randomization, but they
will not appear on any exams.
The derivative of sec x is the product, sec x tan x = (sec x)(tan x),
and not the composite function, sec tan x = sec(tan x).
D.L. White (Kent State University)
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Examples
1
3 sin x
.
Find the derivative of f (x) = √
x + 5x 2
By the quotient rule
d
f 0 (x) =
=
√
2 ) − (3 sin x) d (√x + 5x 2 )
(3
sin
x)
(
x
+
5x
dx
dx
√
( x + 5x 2 )2
√
(3 cos x)( x + 5x 2 ) − (3 sin x)( 21 x −1/2 + 10x)
√
.
( x + 5x 2 )2
D.L. White (Kent State University)
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Examples
2
Find the derivative of F (x) = 3x 5 tan x.
By the product rule, with f (x) = 3x 5 and g (x) = tan x,
d
d
0
5
5
F (x) =
(3x ) (tan x) + (3x )
(tan x)
dx
dx
= 15x 4 tan x + 3x 5 sec2 x.
D.L. White (Kent State University)
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Examples
3
Find the derivative of F (x) = x 2 sin x cos x.
By the general product rule,
with f (x) = x 2 , g (x) = sin x, and h(x) = cos x,
d 2
d
d
F 0 (x) = dx
x sin x cos x + x 2 dx
sin x cos x + x 2 sin x dx
cos x
= 2x sin x cos x + x 2 cos x cos x + x 2 sin x(− sin x).
D.L. White (Kent State University)
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Examples
4
Find the derivative of f (x) =
sec x tan x
.
cos x
By the quotient rule
d
0
f (x) =
=
d
tan x) (cos x) − (sec x tan x) dx
(cos x)
cos2 x
(sec x tan x) tan x + sec x(sec2 x) cos x − (sec x tan x)(− sin x)
.
cos2 x
dx (sec x
D.L. White (Kent State University)
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