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Math 220
October 2
I. Find f 0 (x)
1. f (x) = 6x2 + 5 + 1/x +
√
5
x+
3x+1
√
3x
2. f (x) = (x + 2)3
3. f (x) =
(x+1)2
x4
4. f (x) = ex+3 + ex−3
II. Differentiate
1. f (x) = x2 ex
2. f (x) =
ex
2−ex
√
3. f (x) = (ex + x2 )( x + ex )
4. f (x) =
x+1
x4 +x−1
5. f (x) =
x2 −1
x
6. f (x) =
x2 −1
x+3
7. f (u) =
1
c+u
8. f (x) = x5/3 (x + kex+k )
9. f (x) =
ax+b
cx+d
10. f (x) = (x3 ex )(x5 + 1)(x + 3)
11. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x)
1
√
3
x cos(x)
√
13. f (x) = (3x)2 + 3x
12. f (x) =
14. f (x) = 2 sec(x) − csc(x)
15. f (θ) = eθ (cot(θ) − θ)
16. f (x) = sin(x) cos(x)
17. f (x) = sin(x) cos(x) tan(x)
18. f (x) = sin(x) cos(x)x
19. f (x) =
sin(x) cos(x)
tan(x)
20. f (x) =
sin(x) cos(x)
x
21. f (x) = sec2 (x)
22. f (x) = sin2 (x) + cos2 (x)
23. f (x) =
x2 cos(x) tan(x)
x+sin(x)
24. f (x) =
x+cos(x)
x−sin(x)
Find an equation of the tangent line to the curve at the given point.
1. y = ex cos(x),
2. y = x + tan(x),
(0, 1)
(π, π)
2
I. Find f 0 (x)
√
√
1. f (x) = 6x2 + 5 + 1/x + 5 x + 3x+1
3x
Solution:
f (x) = 6x2 + 5 + x−1 + x1/5 + 3x2/3 + x−1/3
f 0 (x) = 12x + 0 − x−2 + (1/5)x−4/5 + 2x−1/3 + (−1/3)x−4/3
2. f (x) = (x + 2)3
Solution:
f (x) = x3 + 6x2 + 12x + 8
f 0 (x) = 3x2 + 12x + 12
2
3. f (x) = (x+1)
x4
Solution :
2
2
= xx4 + 2x
+ x14 = x−2 + 2x−3 + x−4
f (x) = x +2x+1
x4
x4
f 0 (x) = −2x−3 − 6x−4 − 4x−5
4. f (x) = ex+3 + ex−3
Solution:
f (x) = ex e3 + ex e−3
f 0 (x) = ex e3 + ex e−3 = ex+3 + ex−3
II. Differentiate
1. f (x) = x2 ex
Solution:
f 0 (x) = 2xex + x2 ex
x
e
2. f (x) = 2−e
x
Solution:
3
f 0 (x) =
=
=
=
=
(ex )0 (2 − ex ) − ex (2 − ex )0
(2 − ex )2
ex (2 − ex ) − ex (−ex )
(2 − ex )2
ex (2 − ex ) + ex (ex )
(2 − ex )2
2ex − e2x + e2x
(2 − ex )2
2ex
(2 − ex )2
√
3. f (x) = (ex + x2 )( x + ex )
Solution:
√
√
f 0 (x) = (ex + x2 )0 ( x + ex ) + (ex + x2 )( x + ex )0
√
1
= (ex + 2x)( x + ex ) + (ex + x2 )( √ + ex )
2 x
4. f (x) = x4x+1
+x−1
Solution:
(x + 1)0 (x4 + x − 1) − (x + 1)(x4 + x − 1)0
(x4 + x − 1)2
(x4 + x − 1) − (x + 1)(4x3 + 1)
=
(x4 + x − 1)2
f 0 (x) =
2
5. f (x) = x x−1
Solution:
f (x) =
x2 1
− = x − x−1
x
x
4
f 0 (x) = 1 + x−2
2
−1
6. f (x) = xx+3
Solution:
(x2 − 1)0 (x + 3) − (x2 − 1)(x + 3)0
f (x) =
(x + 3)2
2x(x + 3) − (x2 − 1)
=
(x + 3)2
x2 + 6x + 1
=
(x + 3)2
0
1
7. f (u) = c+u
Solution:
10 (c + u) − 1(c + u)0
(c + u)2
0 − 1(0 + 1)
=
(c + u)2
−1
=
(c + u)2
f 0 (x) =
8. f (x) = x5/3 (x + kex+k )
Solution:
f 0 (x) = (x5/3 )0 (x + kex+k ) + x5/3 (x + kex ek )0
= (5/3)x2/3 (x + kex+k ) + x5/3 (1 + kex ek )
= (5/3)x2/3 (x + kex+k ) + x5/3 (1 + kex+k )
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9. f (x) = ax+b
cx+d
Solution:
(ax + b)0 (cx + d) − (ax + b)(cx + d)0
(cx + d)2
a(cx + d) − (ax + b)c
=
(cx + d)2
acx + ad − acx − bc
=
(cx + d)2
ad − bc
=
(cx + d)2
f 0 (x) =
10. f (x) = (x3 ex )(x5 + 1)(x + 3)
Solution:
f 0 (x) = (x3 ex )0 [(x5 + 1)(x + 3)] + (x3 ex )[(x5 + 1)(x + 3)]0
= (3x2 ex + x3 ex )[(x5 + 1)(x + 3)] + (x3 ex )[(x5 + 1)0 (x + 3) + (x5 + 1)(x + 3)0 ]
= (3x2 ex + x3 ex )[(x5 + 1)(x + 3)] + (x3 ex )[5x4 (x + 3) + (x5 + 1)]
= (3x2 ex + x3 ex )[(x5 + 1)(x + 3)] + (x3 ex )[6x5 + 15x4 + 1)]
11. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x)
Solution:
f 0 (x) = cos(x) − sin(x) + sec2 (x) − csc(x) cot(x) + sec(x) tan(x) − csc2 (x)
√
12. f (x) = 3 x cos(x)
Solution:
6
√
√
f 0 (x) = ( 3 x)0 cos(x) + 3 x(cos(x))0
= (1/3)x−2/3 cos(x) + x1/3 (− sin(x))
= (1/3)x−2/3 cos(x) − x1/3 (sin(x))
13. f (x) = (3x)2 +
Solution:
√
3x
f (x) = 9x2 +
√ √
3 x
√
1
3 √
2 x
√
3
= 18x + √
2 x
f 0 (x) = 18x +
14. f (x) = 2 sec(x) − csc(x)
Solution:
f 0 (x) = 2 sec(x) tan(x) + csc(x) cot(x)
15. f (θ) = eθ (cot(θ) − θ)
Solution:
f 0 (x) = (eθ )0 (cot(θ) − θ) + eθ (cot(θ) − θ)0
= eθ (cot(θ) − θ) + eθ (− csc2 (θ) − 1)
16. f (x) = sin(x) cos(x)
Solution:
7
f 0 (x) = (sin(x))0 cos(x) + sin(x)(cos(x))0
= cos(x) cos(x) − sin(x) sin(x)
= cos2 (x) − sin2 (x)
17. f (x) = sin(x) cos(x) tan(x) Solution:
f (x) = sin(x) cos(x)
sin(x)
= sin(x) sin(x) = sin2 (x)
cos(x)
f 0 (x) = (sin(x))0 sin(x) + sin(x)(sin(x))0
= cos(x) sin(x) + sin(x) cos(x)
= 2 cos(x) sin(x)
18. f (x) = sin(x) cos(x)x
Solution:
f 0 (x) = (sin(x))0 (cos(x)x) + sin(x)(cos(x)x)0
= cos(x)(cos(x)x) + sin(x)(− sin(x)x + cos(x))
= cos2 (x)x − sin2 (x)x + sin(x) cos(x)
19. f (x) =
sin(x) cos(x)
tan(x)
Solution:
f (x) =
sin(x) cos(x)
sin(x)
cos(x)
= cos2 (x)
f 0 (x) = (cos(x))0 cos(x) + cos(x)(cos(x))0
= − sin(x) cos(x) − cos(x) sin(x)
= −2 sin(x) cos(x)
8
20. f (x) =
sin(x) cos(x)
x
Solution:
(sin(x) cos(x))0 x − (sin(x) cos(x))x0
x2
0
[(sin(x)) cos(x) + sin(x)(cos(x))0 ]x − (sin(x) cos(x))
=
x2
2
2
[cos (x) − sin (x)]x − (sin(x) cos(x))
=
x2
f 0 (x) =
21. f (x) = sec2 (x)
Solution:
f (x) = sec(x) sec(x)
f 0 (x) = (sec(x))0 sec(x) + sec(x)(sec(x))0
= (sec(x) tan(x)) sec(x) + sec(x)(sec(x) tan(x))
= 2 sec2 (x) tan(x)
22. f (x) = sin2 (x) + cos2 (x)
Solution:
f 0 (x) = (sin(x))0 sin(x) + sin(x)(sin(x))0 + (cos(x))0 cos(x) + cos(x)(cos(x))0
= cos(x) sin(x) + sin(x) cos(x) − sin(x) cos(x) − cos(x) sin(x)
= cos(x) sin(x) − cos(x) sin(x) + sin(x) cos(x) − sin(x) cos(x)
=0
Alternative solution:
f (x) = sin2 (x) + cos2 (x) = 1
f 0 (x) = 0
9
2
23. f (x) = x
Solution:
f 0 (x) =
=
=
=
=
=
=
cos(x) tan(x)
x+sin(x)
[x2 cos(x) tan(x)]0 (x + sin(x)) − [x2 cos(x) tan(x)](x + sin(x))0
(x + sin(x))2
[2x cos(x) tan(x) + x2 (cos(x) tan(x))0 ](x + sin(x)) − [x2 cos(x) tan(x)](1 − cos(x))
(x + sin(x))2
[2x cos(x) tan(x) + x2 (− sin(x) tan(x) + cos(x) sec2 (x))](x + sin(x))
(x + sin(x))2
[x2 cos(x) tan(x)](1 − cos(x))
−
(x + sin(x))2
[2x cos(x) tan(x) + x2 (− sin2 (x) sec(x) + sec(x))](x + sin(x))
(x + sin(x))2
[x2 cos(x) tan(x)](1 − cos(x))
−
(x + sin(x))2
[2x cos(x) tan(x) + x2 (sec(x)(1 − sin2 (x))](x + sin(x))
(x + sin(x))2
[x2 cos(x) tan(x)](1 − cos(x))
−
(x + sin(x))2
[2x cos(x) tan(x) + x2 (sec(x)(cos2 (x))](x + sin(x))
(x + sin(x))2
[x2 cos(x) tan(x)](1 − cos(x))
−
(x + sin(x))2
[2x cos(x) tan(x) + x2 (cos(x))](x + sin(x)) − [x2 cos(x) tan(x)](1 − cos(x))
(x + sin(x))2
24. f (x) = x+cos(x)
x−sin(x)
Solution:
10
(x + cos(x))0 (x − sin(x)) − (x + cos(x))(x − sin(x))0
(x − sin(x))2
(1 − sin(x))(x − sin(x)) − (x + cos(x))(1 − cos(x))
=
(x − sin(x))2
f 0 (x) = =
Find an equation of the tangent line to the curve at the given point.
1. y = ex cos(x), (0, 1)
Solution:
y 0 = ex cos(x) − ex sin(x)
m = y 0 (0) = e0 cos(0) − e0 sin(0) = 1 − 0 = 1
Tangent line:
(y − 1) = 1(x − 0)
y =x+1
2. y = x + tan(x), (π, π)
Solution:
y 0 = 1 + sec2 (x)
m = y 0 (π) = 1 + sec2 (π) = 1 + 1 = 2
Tangent line:
(y − π) = 2(x − π)
y = 2x − π
11