Derivative Practice 1 (Answers on back)
For problems 1 through 20, find f 0 (x) for the given f (x).
1. f (x) = sin x cos x
2. f (x) =
11. f (x) =
tan x
x
√
sin x +
12. f (x) = √
4
√
3
x
1
x2 + tan x
3x2 + 2
x−7
2
3. f (x) = sin(2x)
13. f (x) =
4. f (x) = 3 csc x + 7 cot x
14. f (x) = x6 + 7x4 + 2x−3 − π
5. f (x) = (x2 + 7)10
15. f (x) =
6. f (x) = 2x cos x
16. f (x) = (x sin x)3
2
tan2 x
(x + 1)5
3
x+1
x−1
7. f (x) = cot(x )
17. f (x) = sin
8. f (x) = sin(cos x)
18. f (x) = csc((x + 1)5 )
9. f (x) = (x3 − 6x)11
19. f (x) = sec(cot(x3 ))
10. f (x) = sin(x2 ) + sec(x2 )
20. f (x) = 7(x3 + 4x)5 sin3 x
For problems 21 through 24, find the indicated derivative of f (x)
21. Find the 3rd derivative of f (x) = 7x4 + 2x3 + x−1 .
22. Find the 4th derivative of f (x) = cos(2x).
23. Find the 3rd derivative of f (x) = x sin x.
24. Find the 2nd derivative of f (x) = tan(x2 ).
For problems 25 through 28, find y 0 for the given y, using implicit differentiation.
25. sin(y 2 ) = 2x.
26. y +
x
x+7
= cos y
27. xy + tan y = csc x
28. 3(y + 2)3 = x3 − 4 sin x − y
29. Find the equation of the tangent line to y = −3(x + 1)3 when x = 1.
30. If an object has position s(t) = 3 sin(2t) at time t, what are its velocity and acceleration when
t = 5π/6?
1
1. − sin2 x + cos2 x
2.
x sec2 x−tan x
x2
3. 2 cos(2x)
4. −3 csc x cot x − 7 csc2 x
5. 10(x2 + 7)9 (2x)
6. 2 cos x − 2x sin x
7. − csc2 (x2 )(2x)
8. cos(cos x) · (− sin x)
9. 11(x3 − 6x)10 (3x2 − 6)
10. 2x cos(x2 ) + 2x sec(x2 ) tan(x2 )
11.
1
−1/2 cos x
2 (sin x)
+ 13 x−2/3
12. − 41 (x2 + tan x)−5/4 (2x + sec2 x)
2 2 +2)
+2
· (x−7)(6x)−(3x
13. 2 3xx−7
(x−7)2
14. 6x5 + 28x3 − 6x−4
15.
(x+1)5 ·2 tan x sec2 x−tan2 x·5(x+1)4
(x+1)10
16. 3(x sin x)2 (sin x + x cos x)
x−1−(x+1)
x+1
17. 3 sin2 x+1
cos
x−1
x−1 ·
(x−1)2
18. − csc((x + 1)5 ) cot((x + 1)5 ) · 5(x + 1)4
19. sec(cot(x3 )) tan(cot(x3 )) · (− csc2 (x3 )) · (3x2 )
20. 21(x3 + 4x)5 sin2 x cos x + 35(x3 + 4x)4 (3x2 + 4) sin3 x
21. f 000 (x) = 168x + 12 − 6x−4
22. f (4) (x) = 16 cos(2x)
23. f 000 (x) = −x cos x − 3 sin x
24. f 00 (x) = 2 sec2 (x2 ) + 8x2 sec2 (x2 ) tan(x2 )
25. y 0 =
1
y cos(y 2 )
26. y 0 =
−7
(x+7)2 (1+sin y)
27. y 0 =
− csc x cot x−y
x+sec2 y
28. y 0 =
3x2 −4 cos x
1+9(y+2)2
29. y + 24 = −36(x − 1)
√
5π
30. v( 5π
)
=
3,
a(
)
=
6
3
6
6
2