Seminar assignments - Practice derivative problems, questions and answers PDF

Preview

Summary

document - document merged files: deriv.pdf - derivsolns.pdf...

Description

Practice Derivative Problems In the following exercise, use the shortcut rules for the derivative to find the desired derivatives. √ 1. f (x) = (2x + 1) x 2 + 1 5 2. f (x) = e ln (x + 6)

√ 1 3. f (x) = 2 x + √ x 4. f (x) = e x + e

x 3

8

5. f (x) = (3x 5 − 1) 4 (x3 + 2) 9 2 6. f (x) = 3x + 5x + 1

7. f (x) = [(x4 − 7x 2)6 + 4x 3] 5 8. f (x) = 1 + ln x + (ln x) 2 + (ln x)3 3

9. f (x) = (x 3 + 5x + 9) 2 ln x ) 10. f (x) = e ln (1 + e r 1 3 11. f (x) = x3 + 3 x 12. f (x) = log3 (log7 (log 5 (x + 2))) 13. f (x) = (ln x + xex + 1) 3 √ 4 14. f (x) = e x + 3x ln (x2 + 2x) 4

15. f (x) = e x +3x

2

+1

(4x3 + 6x) 2

16. f (x) = e e + ln (ln (ln x)) x

ln (x) + 4 4 ] ex 18. f (x) = (x 2 + 6x + 1)4

17. f (x) = [

4

19. f (x) = [ln (x2 + 1)] 3 20. f (x) = e x + ln x + eπ 2 1 4 2 21. f (x) = (x 3 − 3x 2 + 6x− 5 )ex +1 4

2

4

2

22. f (x) = (e x +x + ex + ex )(x4 + x2 ) 23. f (x) = (

3 ln x 3 ) + ex (3x4 + 2x + 1)2 x2 + 1

3 7x4 − x 2 5 ) + (x 2 − 1)3 (2x + x3 )5 ] 8 24. f (x) = [( 6 x q 1 2 25. f (x) = e x + (ex 2 + 1)(ln (x4 + 1) + 3) 2

Practice Derivative Problems Solutions 1

1

1. f ′ (x) = 2(x 2 + 1) 2 + (2x + 1)( 12 )(x2 + 1)− 2 (2x) 2. f ′ (x) =

5x4 ln (x5 +6) e = 5x4 x5 + 6

3. f ′ (x) = x− 12 − 21 x −23 4. f ′ (x) = (1 + e x )ex+e 5. f ′ (x) =

3 (3x5 4

x

8

1

3

1

− 1)− 4 (15x4 )(x3 + 2) 9 + 89 (x3 + 2) − 9 (3x2 )(3x5 − 1) 4

2 +5x+1

6. f ′ (x) = 3 x

(2x + 5)(ln 3)

7. f (x) = 5[(x − 7x 2) 6 + 4x3]4[6(x4 − 7x2 )5 (4x3 − 14x) + 12x2 ] 4

′

8. f ′ (x) =

1 2 ln x 3(ln x)2 + + x x x

9. f ′ (x) =

3 (x3 2

1

+ 5x + 9) 2 (3x2 + 5) ln x

10. f ′ (x) = e ln (1+e

)

1 eln x ( x ln x ) = 1 1+e 2

11. f ′ (x) =

1 (x3 3

12. f ′ (x) =

1 1 1 (log 7 (log5 (x + 2)))(ln 3) (log 5 (x + 2))(ln 7) (x + 2)(ln 5)

+ x−3 ) − 3 (3x2 − 3x−4 )

13. f ′ (x) = 3(ln x + xex + 1) 2( 1x + e x + xex ) √ √ 2x + 2 1 1 4 4 ) 14. f ′ (x) = (x4 + 3x) −2 (4x3 + 3)e x +3x ln (x2 + 2x) + e x +3x ( 2 x + 2x 2

15. f ′ (x) = (4x 3 + 6x)ex x

16. f ′ (x) = e xee + 17. f ′ (x) = 4[

4

+3x2 +1

4

(4x3 + 6x)2 + 2(4x3 + 6x)(12x2 + 6)e x +3x

2

+1

1 1 1 (ln (ln (x))) ln (x) x

ln (x) + 4 3 ex ( x1 ) − ex (ln (x) + 4) ] (ex )2 ex

18. f ′ (x) = 4(x 2 + 6x + 1) 3(2x + 6) 2x 4 1 19. f ′ (x) = [ln (x2 + 1)] 3 2 3 x +1 20. f ′ (x) = e x + x1 2 2 2 1 3 1 24 9 2 4 1 21. f ′ (x) = ( x− 3 − x − 2 − x− 5 )ex +1 + (x 3 − 3x 2 + 6x− 5 )(2xex +1 ) 3 2 5 4

2

4

2

22. f ′ (x) = [(4x3 + 2x)e x +x + (4x3)e x + (2x)e x ](x4 + x2 ) + (ex 23. f ′ (x) = 3(

4

+x2

4

2

+ ex + ex )(4x3 + 2x)

ln (x) 2 (x 2 + 1) 1x − (ln (x))(2x) 3 3 + 3x 2e x (3x4 + 2x + 1)2 + ex (2)(3x4 + 2x + 1)(12x3 + 2) ) [ x2 + 1 (x2 + 1)2

7x4 − x2 4 x6 (28x3 − 2x) − (7x4 − x2 )(6x5 ) 5 3 7x4 − x2 5 ]+ ) [ ) + (x2 − 1)3 (2x + x3 )5 ]− 8 [5( 24. f ′ (x) = [( 6 8 (x6 )2 x6 x 3(x2 − 1)2 (2x)(2x + x3 )5 + (x2 − 1)3 (5)(2x + x3 )4 (2 + 3x2 )] 1 1 2 1 1 1 2 1 2 2 25. f ′ (x) = (e x + (ex + 1)(ln (x4 + 1) + 3)2 )− 2 [2xex + ( x− 2 (ex )(ln (x4 + 1) + 3) 2+ 2 2 1 4x 3 ] (ex 2 + 1)(2)(ln (x4 + 1) + 3) 4 x +1...