Math 220 February 21 I. Find the derivative. 1. f(x)=1/x 2 + 5x3 + πx 2

Math 220
February 21
I. Find the derivative.
1. f (x) = 1/x2 + 5x3 + πx2 + x−
√
2
+ eπ
2. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x) + ex
3. f (x) = 2x + e3x + 32x
4. f (x) = sin(3x) + (x + 1)2 + e4x+1 + tan(cx)
5. f (x) =
√
x2 + 3x + 6
6. f (x) = cos(ex ) + ecos(x)
7. f (x) = sin2 (x)
8. f (x) = e(x
2 +x+x−1 )
9. f (x) = (x2 + x + 3x4 )−2
10. f (x) = tan(3x + sin(x))
11. f (x) =
p
√
sin(x) + sin( x)
12. f (x) = csc(x2 + x)
√
13. f (x) = 4
x+1
1
r
14. f (x) =
15. f (x) =
ex + 1
x2 + 1
x2 + 2
x3 + x
7
√
16. f (x) =
x+3
sin(x + 3)
q
p
√
17. f (x) = x + x + x + 1
18. f (x) = cot(sec(x2 ) + cos(x3 ))
19. f (x) = sin2 (cos2 (x2 + x) + x)
20.
2(
x4
f (x) = 3
II. Find the first and second derivatives.
1. f (x) = ex
2. f (x) = x2 + 2x
3. f (x) = sin2 (x)
4. f (x) = cos(αx) sin(βx)
5. f (x) = x2 ex + xex + ex
2
)
!
1
Solutions
I. Find the derivative.
1. f (x) = 1/x2 + 5x3 + πx2 + x−
Answer:
√
2
+ eπ
√
√
f 0 (x) = −2x−3 + 15x2 + 2πx + (− 2)x− 2−1
2. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x) + ex
Answer:
f 0 (x) = cos(x)−sin(x)+sec2 (x)−csc(x) cot(x)+sec(x) tan(x)−csc2 (x)+ex
3. f (x) = 2x + e3x + 32x
f 0 (x) = ln(2)2x + 3e3x + ln(3)32x 2
4. f (x) = sin(3x) + (x + 1)2 + e4x+1 + tan(cx)
f 0 (x) = 3 cos(3x) + 2(x + 1) + 4e4x+1 + c sec2 (cx)
√
5. f (x) = x2 + 3x + 6
Answer:
1
f 0 (x) = √
(x2 + 3x + 6)0
2
2 x + 3x + 6
1
= √
(2x + 3)
2 x2 + 3x + 6
2x + 3
= √
2 x2 + 3x + 6
6. f (x) = cos(ex ) + ecos(x)
Answer:
3
f 0 (x) = − sin(ex )(ex )0 + ecos(x) (cos(x))0
= − sin(ex )ex − sin(x)ecos(x)
7. f (x) = sin2 (x)
Answer:
f 0 (x) = 2(sin(x))(sin(x))0
= 2 sin(x) cos(x)
= sin(2x)
8. f (x) = e(x
Answer:
2 +x+x−1 )
2 +x+x−1 )
(x2 + x + x−1 )0
2 +x+x−1 )
(2x + 1 − x−2 )
f 0 (x) = e(x
= e(x
9. f (x) = (x2 + x + 3x4 )−2
Answer:
f 0 (x) = −2(x2 + x + 3x4 )−3 (x2 + x + 3x4 )0
= −2(x2 + x + 3x4 )−3 (2x + 1 + 12x3 )
10. f (x) = tan(3x + sin(x))
Answer:
f 0 (x) = sec2 (3x + sin(x))(3x + sin(x))0
= sec2 (3x + sin(x))(ln(3)3x + cos(x))
4
p
√
11. f (x) = sin(x) + sin( x)
Answer:
√ √
1
(sin(x))0 + cos( x)( x)0
f 0 (x) = p
2 sin(x)
√
cos( x)
cos(x)
√
+
= p
2 x
2 sin(x)
12. f (x) = csc(x2 + x)
Answer:
f 0 (x) = − csc(x2 + x) cot(x2 + x)(x2 + x)0
= − csc(x2 + x) cot(x2 + x)(2x + x)
√
13. f (x) = 4
Answer:
x+1
√
( x + 1)0
√
1
= ln(4)4 x+1 ( √ )
2 x
f 0 (x) = ln(4)4
r
14. f (x) =
ex + 1
x2 + 1
Answer:
5
√
x+1
1
ex + 1 0
)
f 0 (x) = r x
( 2
e +1 x +1
2
x2 + 1
x
1
(e + 1)0 (x2 + 1) − (ex + 1)(x2 + 1)0
= r x
(x2 + 1)2
e +1
2
x2 + 1
x 2
e (x + 1) − (ex + 1)2x
1
= r x
(x2 + 1)2
e +1
2
x2 + 1
15. f (x) =
x2 + 2
x3 + x
7
Answer:
0
f (x) = 7
x2 + 2
x3 + x
6 x2 + 2
x3 + x
6 0
(x2 + 2)0 (x3 + x) − (x2 + 2)(x3 + x)0
=7
x3 + x
6 2
(2x)(x3 + x) − (x2 + 2)(3x2 + 1)
x +2
=7
x3 + x
x3 + x
x2 + 2
x3 + x
√
x+3
sin(x + 3)
Answer:
16. f (x) =
√
√
( x + 3)0 sin(x + 3) − ( x + 3)(sin(x + 3))0
f (x) =
sin2 (x + 3)
√
(1/2)(x + 3)−1/2 sin(x + 3) − ( x + 3)(cos(x + 3))
=
sin2 (x + 3)
0
6
q
p
√
17. f (x) = x + x + x + 1
Answer:
0
f (x) =
=
=
q
1
p
(x +
√
2(x + x + x + 1)
x+
√
x + 1)0
1
1+ p
(x +
√
2 x+ x+1
1
p
√
2(x + x + x + 1)
√
!
x + 1)0
!
1
p
√
2(x + x + x + 1)
1
1
1+ p
(1 + √
)
√
2 x+1
2 x+ x+1
18. f (x) = cot(sec(x2 ) + cos(x3 ))
Answer:
f 0 (x) = − csc2 (sec(x2 ) + cos(x3 ))(sec(x2 ) + cos(x3 ))0
= csc2 (sec(x2 ) + cos(x3 ))(sec(x2 ) tan(x2 )2x − sin(x3 )3x2 )
19. f (x) = sin2 (cos2 (x2 + x) + x)
Answer:
f 0 (x) = 2 sin(cos2 (x2 + x) + x) (sin(cos2 (x2 + x) + x))0
= 2 sin(cos2 (x2 + x) + x) (cos(cos2 (x2 + x) + x)(cos2 (x2 + x) + x)0
= 2 sin(cos2 (x2 + x) + x) (cos(cos2 (x2 + x) + x)(2 cos(x2 + x)(cos(x2 + x))0 + 1)
= 2 sin(cos2 (x2 + x) + x) ·
(cos cos2 (x2 + x) + x (2 cos(x2 + x)(− sin(x2 + x)(2x + 1)) + 1)
20.
2(
f (x) = 3
7
x4
)
!
Answer:
2(
x4
f 0 (x) = ln(3)3
!
) x4
2( )
!
x4
2( )
!
0
0
4
ln(2)2(x ) x4
= ln(3)3
= ln(3)3
4
2(x )
4
ln(2)2(x ) 4x3
II. Find the first and second derivatives.
1. f (x) = ex
Answer:
f 0 (x) = ex
f 00 (x) = ex
2. f (x) = x2 + 2x
Answer:
f 0 (x) = 2x + ln(2)2x
f 00 (x) = 2 + (ln(x))2 2x
3. f (x) = sin2 (x)
Answer:
8
f 0 (x) = 2 sin(x) cos(x)
f 00 (x) = 2 cos(x) cos(x) + 2 sin(x)(− sin(x))
= 2(cos( x) − sin2 (x))
= 2 cos(2x)
4. f (x) = cos(αx) sin(βx)
Answer:
f 0 (x) = −α sin(αx) sin(βx) + β cos(αx) cos(βx)
f 00 (x) = −α2 cos(αx) sin(βx) − αβ sin(αx) cos(βx) − βα sin(αx) cos(βx) − β 2 cos(αx) sin(βx)
= −(α2 + β 2 ) cos(αx) sin(βx) − 2αβ sin(αx) cos(βx)
5. f (x) = x2 ex + xex + ex
Answer:
f 0 (x) = 2xex + x2 ex + ex + xex + ex
= x2 ex + 3xex + 2ex
f 00 (x) = 2xex + x2 ex + 3ex + 3xex + 2ex
9