Math 220 February 19 I. Find the derivative: 1. f(x) = π 2. f(x) = x + π

Math 220
February 19
I. Find the derivative:
1. f (x) = π
2. f (x) = x + π 2
3. f (x) = x5 + e5
4. f (x) = ex
5. f (x) =
1
x
6. f (x) =
1
x5
7. f (x) = sin(x)
8. f (x) = cos(x)
9. f (x) =
√
1
1
x+ √
+√
4
3
x
x7
10. f (x) = ex+2 + sin(x + 3)
11. f (u) = 2 sin(u)u2
12. f (x) = x2 ex − 2xex
x2
13. f (x) = x
e +1
14. w(x) = 2 sin(2x)
√
u2 + u3 + eu
√
15. s(u) =
u
16. y = tan(x)
1
17. h(u) = csc(u) + sec(u) + cot(u)
18. f (x) =
sin(x) + cos(x)
x2
19. f (x) =
xex + cos(x)
x sin(x)
20. f (x) = cot(x)(π sin(x) + 52 x2 )
√
x3/2 (ex sin(x) + 2x)
21. f (x) =
x+2
22. f (x) = (x + 1)(x + 2)(x + 3)
23. f (x) = (2x + 1) sin(x)ex
24. f (x) = e3x + sin(3x)
1 − cos2 (x)
sin(x)
√
x sin(x) − x2
26. f (x) =
πex − tan(x)
25. f (x) =
27. f (x) = (x sin(x))2
II. Find an equation to the tangent line to the curve at the given point.
1. y = xex , (0, 0)
2. y = x2 + sin(x), (π, π 2 )
3. y = sin(3x), (π/6, 1)
4. y = Cex , (0, C)
5. y =
3+x
, (2, 1)
1 + x2
2
1
Solutions
I. Find the derivative:
1. f (x) = π
Answer:
f 0 (x) = 0
2. f (x) = x + π 2
Answer:
f 0 (x) = 1
3. f (x) = x5 + e5
Answer:
f 0 (x) = 5x4
4. f (x) = ex
Answer:
f 0 (x) = ex
1
5. f (x) =
x
Answer:
f 0 (x) =
−1
x2
f 0 (x) =
−5
x6
1
6. f (x) = 5
x
Answer:
3
7. f (x) = sin(x)
Answer:
f 0 (x) = cos(x)
8. f (x) = cos(x)
Answer:
f 0 (x) = − sin(x)
√
1
1
+√
x+ √
4
3
x
x7
Answer:
9. f (x) =
f (x) = x1/2 + x−1/3 + x−7/4
1
7
1
f 0 (x) = x−1/2 − x−4/3 − x−11/4
2
3
4
10. f (x) = ex+2 + sin(x + 3)
Answer:
f 0 (x) = ex e2 + sin(x) cos(3) + cos(x) sin(3)
f 0 (x) = ex e2 + cos(x) cos(3) − sin(x) sin(3)
= ex+2 + cos(x + 3)
11. f (u) = 2 sin(u)u2
Answer:
4
f (u) = 2u2 sin(u)
f 0 (u) = 2(u2 )0 sin(u) + 2u2 (sin(u))0
= 2(2u) sin(u) + 2u2 cos(u)
= 4u sin(u) + 2u2 cos(u)
= 2u(2 + u cos(u))
12. f (x) = x2 ex − 2xex
Answer:
f 0 (x) = (x2 )0 ex + x2 (ex )0 − [(2x)0 ex + 2x(ex )0 ]
= 2xex + x2 ex − 2ex − 2xex
x
x
=
2xe
+ x2 ex − 2ex − 2xe
= x2 ex − 2ex
x2
ex + 1
Answer:
13. f (x) =
(x2 )0 (ex + 1) − x2 (ex + 1)0
f (x) =
(ex + 1)2
2x(ex + 1) − x2 ex
=
(ex + 1)2
0
14. w(x) = 2 sin(2x)
Answer:
5
w(x) = 2(2 sin(x) cos(x))
= 4 sin(x) cos(x)
w0 (x) = 4(sin(x))0 cos(x) + 4 sin(x)(cos(x))0
= 4 cos(x) cos(x) + 4 sin(x)(− sin(x))
= 4 cos2 (x) − 4 sin2 (x)
= 4(cos2 (x) − sin2 (x))
= 4 cos(2x)
15. s(u) =
u2 +
√
u3 + eu
√
u
Answer:
u3/2
eu
u2
+
+
u1/2 u1/2 u1/2
eu
= u3/2 + u + 1/2
u
s(u) =
3 1/2
(eu )0 u1/2 − eu (u1/2 )0
ds
0
= s (u) = u + 1 +
du
2
(u1/2 )2
eu u1/2 − eu ( 12 u−1/2 )
3
= u1/2 + 1 +
2
u
3 1/2
eu
eu
= u + 1 + 1/2 − 3/2
2
u
2u
16. y = tan(x)
Answer:
6
y=
sin(x)
cos(x)
dy
(sin(x))0 cos(x) − sin(x)(cos(x))0
= y0 =
dx
cos2 (x)
cos(x) cos(x) − sin(x)(− sin(x))
=
cos2 (x)
cos2 (x) + sin2 (x)
=
cos2 (x)
1
=
cos2 (x)
= sec2 (x)
17. h(u) = csc(u) + sec(u) + cot(u)
Answer:
dh
= h0 (u) = − csc(u) cot(u) + sec(u) tan(u) − csc2 (u)
du
sin(x) + cos(x)
18. f (x) =
x2
Answer:
(sin(x) + cos(x))0 x2 − (sin(x) + cos(x))(x2 )0
(x2 )2
(cos(x) − sin(x))x2 − (sin(x) + cos(x))2x
=
(x4 )
(x2 − 2x) cos(x) − (x2 + 2x) sin(x)
=
(x4 )
f 0 (x) =
7
xex + cos(x)
x sin(x)
Answer:
19. f (x) =
f 0 (x) =
=
=
=
=
=
=
=
=
=
[xex + cos(x)]0 x sin(x) − [xex + cos(x)](x sin(x))0
(x sin(x))2
[(xex )0 − sin(x)]x sin(x) − [xex + cos(x)](sin(x) + x cos(x))
x2 sin2 (x)
[(ex + xex ) − sin(x)]x sin(x) − [xex + cos(x)](sin(x) + x cos(x))
x2 sin2 (x)
[x sin(x)ex + x2 sin(x)ex − x sin2 (x)] − [x sin(x)ex + x2 cos(x)ex + cos(x) sin(x) + x co
x2 sin2 (x)
2
x
2
x
x
2
x
[
x sin(x)e
x sin(x)e
+ x sin(x)e − x sin (x)] − [
+ x cos(x)e + cos(x) sin(x) + x co
x2 sin2 (x)
[x2 sin(x)ex − x sin2 (x)] − [x2 cos(x)ex + cos(x) sin(x) + x cos2 (x)]
x2 sin2 (x)
x2 sin(x)ex − x sin2 (x) − x2 cos(x)ex − cos(x) sin(x) − x cos2 (x)
x2 sin2 (x)
x2 sin(x)ex − x2 cos(x)ex − x sin2 (x) − x cos2 (x) − cos(x) sin(x)
x2 sin2 (x)
x2 ex (sin(x) − cos(x)) − x(sin2 (x) + cos2 (x)) − cos(x) sin(x)
x2 sin2 (x)
x2 ex (sin(x) − cos(x)) − x − cos(x) sin(x)
x2 sin2 (x)
20. f (x) = cot(x)(π sin(x) + 52 x2 )
Answer:
f 0 (x) = (cot(x))0 (π sin(x) + 52 x2 ) + cot(x)(π sin(x) + 52 x2 )0
= − csc2 (x)(π sin(x) + 52 x2 ) + cot(x)(π cos(x) + 52 2x)
= − csc2 (x)(π sin(x) + 25x2 ) + cot(x)(π cos(x) + 50x)
8
x3/2 (ex sin(x) +
21. f (x) =
x+2
Answer:
√
2x)
√
0
3/2 x
2x)]
(x
+
2)
−
[x
(e
sin(x)
+
2x)](x + 2)0
f 0 (x) =
(x + 2)2
√ √
√
[(x3/2 )0 (ex sin(x) + 2x) + x3/2 (ex sin(x) + 2 x)0 ](x + 2)
=
(x + 2)2
√
[x3/2 (ex sin(x) + 2x)]1
−
(x + 2)2
√
√
[ 32 x1/2 (ex sin(x) + 2x) + x3/2 (ex sin(x) + ex cos(x) + 2√2x )](x + 2)
=
(x + 2)2
√
x3/2 (ex sin(x) + 2x)
−
(x + 2)2
√
√
√
[ 23 x(ex sin(x) + 2x) + x3 (ex sin(x) + ex cos(x) + √12x )](x + 2)
=
(x + 2)2
√
√
x3 (ex sin(x) + 2x)
−
(x + 2)2
[x3/2 (ex sin(x) +
√
22. f (x) = (x + 1)(x + 2)(x + 3)
Answer:
f 0 (x) = (x + 1)0 (x + 2)(x + 3) + (x + 1)(x + 2)0 (x + 3) + (x + 1)(x + 2)(x + 3)0
= 1(x + 2)(x + 3) + (x + 1)1(x + 3) + (x + 1)(x + 2)1
= (x + 2)(x + 3) + (x + 1)(x + 3) + (x + 1)(x + 2)
= (x + 2 + x + 1)(x + 3) + x2 + 3x + 2
= (2x + 3)(x + 3) + x2 + 3x + 2
= 2x2 + 9x + 9 + x2 + 3x + 2
= 3x2 + 12x + 11
9
23. f (x) = (2x + 1) sin(x)ex
Answer:
f 0 (x) = (2x + 1)0 sin(x)ex + (2x + 1)(sin(x))0 ex + (2x + 1) sin(x)(ex )0
= 2 sin(x)ex + (2x + 1) cos(x)ex + (2x + 1) sin(x)ex
= ex [(2x + 3) sin(x) + (2x + 1) cos(x)]
24. f (x) = e3x + sin(3x) Answer:
10
f (x) = ex ex ex + sin(x) cos(2x) + cos(x) sin(2x)
= ex ex ex + sin(x)(cos(x) cos(x) − sin(x) sin(x)) + cos(x)2 sin(x) cos(x)
= ex ex ex + sin(x) cos(x) cos(x) − sin(x) sin(x) sin(x)) + 2 sin(x) cos(x) cos(x)
f 0 (x) = (ex )0 ex ex + ex (ex )0 ex + ex ex (ex )0
+ (sin(x))0 cos(x) cos(x) + sin(x)(cos(x))0 cos(x) + sin(x) cos(x)(cos(x))0
− (sin(x))0 sin(x) sin(x) − sin(x)(sin(x))0 sin(x) − sin(x) sin(x)(sin(x))0
+ 2(sin(x))0 cos(x) cos(x) + 2 sin(x)(cos(x))0 cos(x) + 2 sin(x) cos(x)(cos(x))0
= 3(ex )0 ex ex
+ (sin(x))0 cos(x) cos(x) + 2 sin(x)(cos(x))0 cos(x)
− 3(sin(x))0 sin(x) sin(x)
+ 2(sin(x))0 cos(x) cos(x) + 4 sin(x)(cos(x))0 cos(x)
= 3ex ex ex
+ cos(x) cos(x) cos(x) − 2 sin(x) sin(x) cos(x)
− 3 cos(x)) sin(x) sin(x)
+ 2 cos(x) cos(x) cos(x) − 4 sin(x) sin(x) cos(x)
= 3e3x + 3 cos(x) cos(x) cos(x) − 9 sin(x) sin(x) cos(x)
= 3e3x + 3 cos(x) cos(x) cos(x) − 3 sin(x) sin(x) cos(x) − 6 sin(x) sin(x) cos(x)
= 3e3x + 3 cos(x)(cos2 (x) − sin(x)2 ) − 3 sin(x)(2 sin(x) cos(x))
= 3e3x + 3 cos(x)(cos(2x)) − 3 sin(x)(sin(2x))
= 3e3x + 3 cos(x + 2x)
= 3e3x + 3 cos(3x)
1 − cos2 (x)
sin(x)
Answer:
25. f (x) =
11
sin2 (x)
sin(x)
= sin(x)
f (x) =
f 0 (x) = cos(x)
√
26. f (x) =
x sin(x) − x2
πex − tan(x)
Answer:
√
√
( x sin(x) − x2 )0 (πex − tan(x)) − ( x sin(x) − x2 )(πex − tan(x))0
f (x) =
(πex − tan(x))2
√
√
x
√ +
( sin(x)
x
cos(x)
−
2x)(πe
−
tan(x))
−
(
x sin(x) − x2 )(πex − sec2 (x))
2 x
=
(πex − tan(x))2
0
27. f (x) = (x sin(x))2
Answer:
f (x) = x sin(x)x sin(x)
= x2 sin(x) sin(x)
f 0 (x) = (x2 )0 sin(x) sin(x) + x2 (sin(x))0 sin(x) + x2 sin(x)(sin(x))0
= 2x sin(x) sin(x) + 2x2 cos(x) sin(x)
= 2x sin(x) (sin(x) + x cos(x))
II. Find an equation to the tangent line to the curve at the given point.
1. y = xex , (0, 0)
Answer:
12
y 0 = ex + xex
m = y 0 (0) = e0 + 0 = 1
Tangent line: y = x
2. y = x2 + sin(x), (π, π 2 )
Answer:
y 0 = 2x + cos(x)
m = y 0 (π) = 2π + cos(π) = 2π − 1
Tangent line: (y − π 2 ) = (2π − 1)(x − π)
3. y = sin(3x), (π/6, 1)
Answer:
y 0 = 3 cos(3x)
m = y 0 (π/6) = 3 cos(3π/6) = 3(0) = 0
Tangent line: (y − 1) = 0(x − π/6) =⇒ y = 1
4. y = Cex , (0, C)
Answer:
13
y 0 = Cex
m = y 0 (0) = Ce0 = C
Tangent line: (y − C) = C(x − 0)
3+x
, (2, 1)
1 + x2
Answer:
5. y =
(3 + x)0 (1 + x2 ) − (3 + x)(1 + x2 )0
y =
(1 + x2 )2
(1 + x2 ) − (3 + x)2x
=
(1 + x2 )2
1 − 6x − x2
=
(1 + x2 )2
0
m = y 0 (2) =
1 − 12 − 4
−15
−3
=
=
2
5
25
5
Tangent line: (y − 1) = − 53 (x − 2)
14