mat 270 - derivative practice 1rhs.soowook.com/derivative practices w solutions.pdfsin 7 1 = − 39....
TRANSCRIPT
MAT 270 - Derivative Practice I Find the derivative of each of the following functions and simplify. 1. π−+−= xxxxf 234)( 23
2. 2
2 33
)(x
xxf −=
3. ( )1523)( 2 +−−= xxxf
4. x
xxf 1)( −=
5. 21)(
−+
=xxxf
6. 2
2 2)(xxxf −
=
7. 2
)( 2
2
−=xxxf
8. ( )1)( 2 += xxxf
9. 1
)(−
= x
x
eexf
10. 2
2)( xx
xf +=
11. 1
2)(−
=xxxf
12. ( )( )1223)( +−= xxxf
13. x
xxy 355 2 −−=
14. 1−
=xxy
15. xeyx
=
16. 176 51
23
++=−
xxy
17. 317x
y−−
=
18.
−
=π
43
34 xy
19. x
y71
=
20.
−
=e
xy 21
2 Bonus:
7ln 32 −−= xey x
MAT 270 - Derivative Practice II Find the derivative of the following functions. 1. ( ) ( )52 43 −= xxf 2. ( ) ( )xxxf 32 23= 3. ( ) ( )312 43 += − xexf x
4. ( )( )312
2
−=
xexgx
5. ( ) ( ) ( )422 23 xxxxexg x +−++=
6. ( ) ( )xxxf
532 52−
=
7. ( )xy 3cos=
8. 2
sin1cos
−
=xxy
9. ( )502 517 xxy −= 10. ( )( )xey x 3sin2= 11. xy sin=
12. 1
tan2 −
=x
xy
13. ( )2arcsin xy = 14. ( ) ( )xxy arctan12 +=
15. ( )[ ]3arccos xy = 16. ( )xy 6tan=
17. xxy
2cos2sin
=
18. 2
sinxxy =
19. ( )π1sintan += xy
20. ( ) ( )9sin35cos3 xxy += 21. ( )123sin 23 +−= xxy
22.
=x
xy 1tan2
23. ( ) ( )xxf 2sin= 24. ( ) ( )xexg x 2cos3= 25. ( )[ ]43arcsin xy = 26. ( )16tan 2 −= xy 27. ( ) xey 3sin=
28. 3
22 tansecx
xxy −=
29. 3
cosxxy =
30. ( )( )e
xy 14sinsin +=
31. ( )xxy 73cos 22 −=
32.
=x
xy 1sin3
33. ( )xy 4cos=
34. 12
tan−
=xxy
35. 3 1sin −= xy
36. ( ) 23sin π+= xexy
37. xx eey −+=
π
38. xxy cos61sin
71
−=
39. x
xxy22 cotcsc −
=
40. ( )( )xxy
9sin9cos
=
41. ( )371tansin += xy
42.
−=x
xy 1tan4 5
Derivative Practice III Find the derivative of each of the following functions. 1. 22 2 π+= xxy 2. ( )2arcsin xy = 3. xy −= 510 4. ( )[ ]3arccos xy = 5. ( )xey arctan=
6. ( ) xx
xxf −⋅=
2
34
7. ( ) 735 xxg x += 8. ( ) ( )xxf 5arctan −= 9. yxy sin2 2 += 10. ( )3arccos xy = 11. ( )[ ]4arcsin xy = 12. ( ) ( )xxf 2arctan −= 13. yxy cos3 3 +=
14. ( )xey x 20csc 110 −= 15. ( )xy 7sec 1−= 16. 1coscos =+ xyyx
17. 12 +=−
xyxy
18. yxyyx 43 232 −=+ 19. xyyxxy =−+− 11
20. ( )23
222 yxxy +=
MAT 270 - Derivative Practice I Solutions 1. π−+−= xxxxf 234)( 23
2612)(' 2 +−= xxxf
2. 2
2 33
)(x
xxf −=
3
632)('
xxxf +=
3. ( )1523)( 2 +−−= xxxf
1512)(' += xxf
4. x
xxf 1)( −=
xxxxf
21
21)(' +=
5. 21)(
−+
=xxxf
2)2(3)('
−−
=x
xf
6. 2
2 2)(xxxf −
=
3
4)('x
xf =
7. 2
)( 2
2
−=xxxf
22 )2(4)('−
−=x
xxf
8. ( )1)( 2 += xxxf
xxxxf
21
25)(' +=
9. 1
)(−
= x
x
eexf
( )21)('
−
−=
x
x
eexf
10. 2
2)( xx
xf +=
xxxxf
411)(' +
−=
11. 1
2)(−
=xxxf
2)1(2)('
−−
=x
xf
12. ( )( )1223)( +−= xxxf
112)(' −= xxf
13. x
xxy 355 2 −−=
2
32
510'xx
xy +−=
14. 1−
=xxy
( )212
1'−
−=
xxy
15. xeyx
=
2'xe
xey
xx
−=
16. 176 51
23
++=−
xxy
−
−
+−= 54
25
579' xxy
17. 317x
y−−
=
( )23
2
121'xxy
−
−=
18.
−
=π
43
34 xy
−−
−
ππ 4
1
341 x
19. x
y71
=
271'x
y −=
20.
−
=e
xy 21
2
−−
−=e
xey 21
)21(' Bonus:
7ln 32 −−= xey x
8
ln 212'2
xxey
x
+=
MAT 270 - Derivative Practice II Find the derivative of the following functions. 1. ( ) ( )52 43 −= xxf
( ) ( ) ( ) ( )4242 43306435 −=−=′ xxxxxf 2. ( ) ( )xxxf 32 23=
( ) ( ) ( )( )2ln2926 323 xx xxxf +=′ 3. ( ) ( )312 43 += − xexf x
( ) ( ) ( )( ) ( ) ( ) ( )312212312212 4324394323433 +++=+++=′ −−−− xexexexexf xxxx
4. ( )( )312
2
−=
xexgx
( ) ( ) ( )( )6
23
12126122
22
−−−−
=′x
xexxexgxx
5. ( ) ( ) ( )422 23 xxxxexg x +−++= ( ) ( ) ( ) ( ) ( )162341212623412 322322 −+−++=+−+−++=′ xxxxexxxxexg xx
6. ( ) ( )xxxf
532 52−
=
( ) ( ) ( )( ) ( ) ( ) ( )2
54
2
54
253253275
2532553325
xxxx
xxxxxf −−−−
=−−−−
=′
7. ( )xy 3cos=
xxxy
2sincos3 2
−=′
8. 2
sin1cos
−
=xxy
( )2sin1cos2xxy
−=′
9. ( )502 517 xxy −=
( ) ( )53451750 492 −−=′ xxxy 10. ( )( )xey x 3sin2=
xexey xx 3cos33sin2 22 +=′ 11. xy sin=
xxy
sin2cos
=′
12. 1
tan2 −
=x
xy
( )( )22
22
1tan2sec1
−
−−=′
xxxxxy
13. ( )2arcsin xy =
412xxy−
=′
14. ( ) ( )xxy arctan12 +=
1arctan2 +=′ xxy 15. ( )[ ]3arccos xy =
( )2
2
1arccos3
xxy
−−=′
16. ( )xy 6tan=
xy 6sec6 2=′
17. xxy
2cos2sin
=
xy 2sec2 2=′
18. 2
sinxxy =
3
sin2cosx
xxxy −=′
19. ( )π1sintan += xy
( )xxy sinseccos 2=′ 20. ( ) ( )9sin35cos3 xxy +=
98 cos275sin15 xxxy +−=′ 21. ( )123sin 23 +−= xxy
( ) ( )( )26123cos123sin3 222 −+−+−=′ xxxxxy
22.
=x
xy 1tan2
−
=′
xxxy 1sec1tan2 2
23. ( ) ( )xxf 2sin=
( )x
xxxf cossin=′
24. ( ) ( )xexg x 2cos3= ( ) ( )xxexg x 2sin22cos33 −=′
25. ( )[ ]43arcsin xy =
( )[ ]( )
( )2
23
33 31
1arcsin4 xx
xy
−=′
26. ( )16tan 2 −= xy
( )16sec12 22 −=′ xxy 27. ( ) xey 3sin=
3sinxey =′
28. 3
22 tansecx
xxy −=
43 −−=′ xy
29. 3
cosxxy =
4
cos3sinx
xxxy +−=′
30. ( )( )e
xy 14sinsin +=
( )[ ]xxy 4sincos4cos4=′
31. ( )xxy 73cos 22 −=
( ) ( ) ( )xxxxxy 73sin73cos1214 22 −−−=′
32.
=x
xy 1sin3
−
=′
xx
xxy 1cos1sin3 2
33. ( )xy 4cos=
xxxy sincos2 3
−=′
34. 12
tan−
=xxy
( )( )2
2
12tan2sec12
−−−
=′x
xxxy
35. 3 1sin −= xy
( )3 21sin3
cos
−=′
x
xy
36. ( ) 23sin π+= xexy
( )xxey x cossin +=′
37. xx eey −+=
π
( )( )2xx
xx
eeeey−
−
+
−−=′π
38. xxy cos61sin
71
−=
xxy sin61cos
71
+=′
39. x
xxy22 cotcsc −
=
2
1x
y −=′
40. ( )( )xxy
9sin9cos
=
( )xy 9csc9 2−=′
41. ( )371tansin += xy
( )xxy tancossec2=′
42.
−=x
xy 1tan4 5
−−
−=′
xx
xxy 1sec41tan20 234
Derivative Practice III Find the derivative of each of the following functions. 1. 22 2 π+= xxy
( )2ln22 2xxy x +=′ 2. ( )2arcsin xy =
412xxy−
=′
3. xy −= 510
( )( ) ( )( )110ln1021
21
5 −=′−
−xy
4. ( )[ ]3arccos xy =
( )2
2
1arccos3
xxy
−
−=′
5. ( )xey arctan=
x
x
eey 21+
=′
6. ( ) xx
xxf −⋅=
2
34
( ) ( )( ) ( )( )( )( )123ln343422 12 −+−=′ −−−− xxxxf xxxx
7. ( ) 735 xxg x += ( ) 6215ln5 xxg x +=′
8. ( ) ( )xxf 5arctan −=
( ) 22515x
xf+−
=′
9. yxy sin2 2 +=
yx
dxdy
cos22−
=
10. ( )3arccos xy =
6
2
13xxy−
−=′
11. ( )[ ]4arcsin xy =
( )2
3
1arcsin4
xxy
−=′
12. ( ) ( )xxf 2arctan −=
2412x
y+−
=′
13. yxy cos3 3 +=
yx
dxdy
sin33 2
+=
14. ( )xey x 20csc 110 −=
( )( )( )
−
−+=′ −
12020
12020csc102
10110
xxexey xx
15. ( )xy 7sec 1−=
( ) 1491
177
722 −
=−
=′xxxx
y
16. 1coscos =+ xyyx
yxxyxy
dxdy
sincoscossin
−−
=
17. 12 +=−
xyxy
2123
2
2
++−
=x
xyxdxdy
18. yxyyx 43 232 −=+
46321
22
3
++−
=yyx
xydxdy
19. xyyxxy =−+− 11
( ) ( )
( ) ( ) xyxx
xyyy
dxdy
−−+−
−−−−=
−
−
21
21
21
21
12
1
12
1
20. ( )23
222 yxxy +=
( )( )2
122
21
22
32
23
yxyx
yyxxdxdy
+−
−+=