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Simple Derivative Rules: Finding Derivatives of Functions - Prof. John H. Peacher-Ryan, Study notes of Calculus

Christian Brothers University (CBU)Calculus

Prof. John H. Peacher-Ryan

The simple derivative rules to calculate the derivatives of various functions. The rules include the constant rule, linear function rule, power rule, exponential rule, natural exponential rule, natural logarithm rule, constant multiplier rule, sum rule, and more. Use these rules to find derivatives of given functions.

Typology: Study notes

Pre 2010

Uploaded on 08/13/2009

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Math 131 (Fall 2008) Simple Derivative Rules

Rule Name Function Derivative

Constant Rule

y b

y0

Linear Function Rule

y ax b 

y a

Power Rule

y x



, where n is any real

number





y n x

(for those values of x for

which both

and

n x

n1

are defined)

Exponential Rule

y b



, where

b0

y b b

Natural Exponential Rule

y e



y e

Natural Logarithm Rule

y xln

, where

x0

yx

Constant Multiplier Rule

 

y k f x

, where k is a constant

 



y k f x

Sum Rule

   

y f x g x 

   



y f x g x

Use the Simple Derivative Rules to find derivatives of the following functions.

y x  3

y x 5 3 14.

(

y5

)

 

f x x 

73 14.

Discover Study notes of Calculus Christian Brothers University (CBU)

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Download Simple Derivative Rules: Finding Derivatives of Functions - Prof. John H. Peacher-Ryan and more Study notes Calculus in PDF only on Docsity!

Math 131 (Fall 2008) Simple Derivative Rules

Rule Name Function Derivative

Constant Rule y^ ^ b y^ ^ ^0

Linear Function Rule y^  ax^^ ^ b y^ ^  a

Power Rule y^^  x^^ n , where n is any real

number

y   n x n^ ^1

(for those values of x for

which both x n^ and n^ x^

n  1

are defined)

Exponential Rule y  b x , where b  0 y   b x^ ln b

Natural Exponential Rule y^^  e^ x y^ ^  ex

Natural Logarithm Rule y^ ln^ x , where x  0 y^ ^  x

Constant Multiplier Rule (^) y  k f  (^) x , where k is a constant y   k f  (^) x  Sum Rule (^) y  f  (^) x  (^)  g  (^) x  y   f  x  (^)  g  (^) x 

Use the Simple Derivative Rules to find derivatives of the following functions.

1. y^ ^ x ^3

2. y^ ^5 x ^3.^14 ( y^ ^ ^5 )

f  x  x 

4. y 

5. f ^ x   4. 57 ( f  x  0 )

6. y^ ^4 x^^ ^8 x

7. y  2. 5 x 2  3 (

dy

dx

 5 x )

8. y  5 x  3 x 

9. y  2. 5 x^10

10. y  3 x ^8

11. y  2. 5 x^4  5. 7 x  12. 1 ( y   10 x 3  5. 7 )

12. y^ ^3 x

13. y^ ^5 x^ ^2 x ( y^ ^ ^ 

14. y

x

15. f  x 

x

3 (^ ^ ^4

x

f ^ x  )

16. y x

x

7 (^ ^ ^ ^

y 14 x^6 21 x ^8

17. y  2. 3 e x

18. y  3 e  5 e x

19. y e

x x

. (^ y^ ^ ^ e^  x

2 3 x^2

20. y^ ^1 ^ x^ ^5 e

2 3

( y^ ^ ^2 x )

21. y^

 5 x

22. y

23. y^^ ^3 x^  x^3 ( y^ ^ ^3 x^ ln^3 ^3 x^2 )

24. y^ ^2.^01 x^^^5 ^4.^7 x^^3 ^7 x

25. y x

x

 2   ex

ex

x

y x 7

26. y x x

27. y

x

( y  

x

28. y^ ln x

29. y^ ^4 ln^ x ( y  

x

30. g ^ x  3 x^5  2 ln x  5

31. y^ e^ x

 3 x ln

( dy^ dx^ e^ x

 3 x  ^1

32. h ^^ x^ ^ ^ 

 15 5. 25 x

( h  x ^  ^ 

15 5. 25 ln 5. 25 )

33. y  x  x 

2 ln ln 4

34. y

Since y   ^ 

x (^) x

y   1500 ln  1. 26824   1. 26824  x.

35. y

36. y^^ ^5.^32 e^^ x^ ^9 x ^7 x^ ^3

37. h  x  x

x

 2   x

38. y^ ^5 ^9 x

2 3

Simple Derivative Rules: Finding Derivatives of Functions - Prof. John H. Peacher-Ryan, Study notes of Calculus

Related documents

Partial preview of the text

Download Simple Derivative Rules: Finding Derivatives of Functions - Prof. John H. Peacher-Ryan and more Study notes Calculus in PDF only on Docsity!

Math 131 (Fall 2008) Simple Derivative Rules

Rule Name Function Derivative

Constant Rule y^ ^ b y^ ^ ^0

Linear Function Rule y^  ax^^ ^ b y^ ^  a

Power Rule y^^  x^^ n , where n is any real

number

(for those values of x for

which both x n^ and n^ x^

are defined)

Exponential Rule y  b x , where b  0 y   b x^ ln b

Natural Exponential Rule y^^  e^ x y^ ^  ex

Natural Logarithm Rule y^ ln^ x , where x  0 y^ ^  x

Use the Simple Derivative Rules to find derivatives of the following functions.

1. y^ ^ x ^3

2. y^ ^5 x ^3.^14 ( y^ ^ ^5 )

4. y 

5. f ^ x   4. 57 ( f  x  0 )

6. y^ ^4 x^^ ^8 x

7. y  2. 5 x 2  3 (

dy

dx

 5 x )

8. y  5 x  3 x 

9. y  2. 5 x^10

10. y  3 x ^8

11. y  2. 5 x^4  5. 7 x  12. 1 ( y   10 x 3  5. 7 )

12. y^ ^3 x

13. y^ ^5 x^ ^2 x ( y^ ^ ^ 

14. y

x

15. f  x 

x

3 (^ ^ ^4

x

f ^ x  )

16. y x

x

7 (^ ^ ^ ^

17. y  2. 3 e x

18. y  3 e  5 e x

19. y e

x x

. (^ y^ ^ ^ e^  x

2 3 x^2

20. y^ ^1 ^ x^ ^5 e

( y^ ^ ^2 x )

21. y^

22. y

23. y^^ ^3 x^  x^3 ( y^ ^ ^3 x^ ln^3 ^3 x^2 )

24. y^ ^2.^01 x^^^5 ^4.^7 x^^3 ^7 x

25. y x

x

 2   ex

ex

x

y x 7

26. y x x

27. y

x

x

( y  

x

28. y^ ln x

29. y^ ^4 ln^ x ( y  

x

30. g ^ x  3 x^5  2 ln x  5

31. y^ e^ x

( dy^ dx^ e^ x

32. h ^^ x^ ^ ^ 

( h  x ^  ^ 

15 5. 25 ln 5. 25 )

33. y  x  x 

2 ln ln 4

34. y

Since y   ^ 

y   1500 ln  1. 26824   1. 26824  x.