Chapter 3.7: Derivatives of the Trigonometric Functions
Chapter 3.7: Derivatives of the Trigonometric Functions
Expected Skills:
? Know (and be able to derive) the derivatives of the 6 elementary trigonometric functions.
? Be able to use the product, quotient, and chain rules (where appropriate) to differentiate functions involving trigonometry.
? Be able to use the derivative to calculate the instantaneous rates of change of a trigonometric function at a given point.
? Be able to use the derivative to calculate the slope of the tangent line to the graph of a trigonometric function at a given point.
? Be able to use the derivative to calculate to answer other application questions, such as local max/min or absolute max/min problems.
Practice Problems:
1. Fill in the given table:
f (x) sin x cos x tan x cot x sec x csc x
f (x)
f (x) sin x cos x tan x cot x sec x csc x
f (x) cos x - sin x sec2 x - csc2 x sec x tan x - csc x cot x
1
d 2. Use the definition of the derivative to show that (cos x) = - sin x
dx
d
cos (x + h) - cos x
(cos x) = lim
dx
h0
h
cos x cos h - sin x sin h - cos x
= lim
h0
h
cos x cos h - cos x sin x sin h
= lim
-
h0
h
h
cos h - 1
sin h
= lim cos x
- sin x
h0
h
h
= (cos x)(0) - (sin x)(1)
= - sin x
3. Use the quotient rule to show that d (cot x) = - csc2 x. dx
d
d cos x
(cot x) =
dx
dx sin x
(sin x)(- sin x) - (cos x)(cos x)
=
sin2 x
-(sin2 x + cos2 x)
=
sin2 x
1 = - sin2 x
= - csc2 x
d 4. Use the quotient rule to show that (csc x) = - csc x cot x.
dx
d
d1
(csc x) =
dx
dx sin x
(sin x)(0) - (1)(cos x)
=
sin2 x
cos x
= - sin2 x
1 cos x =-
sin x sin x
= - csc x cot x
2
tan 5. Evaluate lim
3
+
h
- tan
3
by interpreting the limit as the derivative of a
h0
h
function at a particular point.
tan lim
3
+
h
- tan
3
d = (tan x)
= sec2 = 4
h0
h
dx
x=
3
3
For problems 6-16, differentiate
6. f (x) = 2 cos x + 4 sin x -2 sin x + 4 cos x sin2 x
7. f (x) = cos x
2 sin x + sin x tan2 x 8. f (x) = x3 sin x
3x2 sin x + x3 cos x 9. f (x) = sec2 x + tan2 x
4 sec2 (x) tan (x)
1 10. f (x) = tan
x2
-2x-3 sec2
1 x2
11. f (x) = sec 2x 2 sec (2x) tan (2x)
12. f (x) = cos3 3x -9 sin (3x) cos2 (3x)
13. f (x) = sin x
-x-2 cos x
14. f (x) = sin (sin 2x) 2 cos (sin 2x) cos 2x
3
15. f (x) = tan2 (x2 - 1) 4x tan (x2 - 1) sec2 (x2 - 1)
16. f (x) = 4x2 csc 5x 8x csc (5x) - 20x2 csc (5x) cot (5x)
d
17. Use the following table to calculate g 2 sin x
dx
4
x=3
x f (x) f (x) g(x) g (x) 1 -2 -5 3 9 2 5 -3 4 -2 3 -1 6 7 -6 4 3 1 -2 5 54 7 1 8
9 -
4
18. What is the 100th derivative of y = sin (2x)?
2100 sin 2x
cos x
19. Compute an equation of the line which is tangent to the graph of f (x) =
at the
x
point where x = .
12 y= x-
2
20. Find all points on the graph of y = sin2 x where the tangent lines are parallel to the line y = x. + k where k is any integer 4
For problems 21-22, find all values of x in the interval [0, 2] where the graph of the given function has horizontal tangent lines.
21. f (x) = sin x cos x 3 5 7 ,,, 44 4 4
22. g(x) = csc x 3 , 22
4
23. Use the Intermediate Value Theorem to show that there is at least one point in the
interval (0, 1) where the graph of f (x) = sin x - 1 x3 will have a horizontal tangent
3
line.
with plots
animate, animate3d, animatecurve, arrow, changecoords, complexplot, complexplot3d,
(1)
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least
one
x0
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where f (x) will have seatophtioonrs3idz,ospnacteaculrvte,aspnagrseemnattrixlpilnote, s.urfdata, textplot, textplot3d, tubeplot
a d plot
sqrt
2
sin
x
,
x
=
0 ..
Pi 2
,
scaling
=
constrained
24.
Consider the
interval 0,
graphs of f (x) b d plot sqrt 2
=
cos
x
, x2=c0o.. sP(i x, s)caliangPnL=dOcoTngs..t(.raxin)ed=
.
2
2 sin(x) shown belo(w2) on the
2
PLOT ...
(3)
display a, b
1.4
1.2
1
0.8
0.6
0.4
0.2
0
3 5 3 7
16
8 16
4
16
8 16
2
x
Show that the graphs of f (x) and g(x) intersect at a right angle when x = . (Hint:
4
Show that the tangent lines to f and g at x = are perpendicular to each other.)
4
f
= -1 and g
= 1. So, the tangent lines to f and g at x = are
4
4
4
perpendicular to one another since the product of their slopes is -1.
5
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