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.

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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

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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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