Calculus AB, BC and BCD Drill on Derivatives

1
Calculus AB, BC and BCDDrill on
Derivatives!!!!!!!!!!

• By Susan E. Cantey
• Walnut Hills H.S.
• 2006

2
Derivative Drill Practice

• Im going to ask you to remember some
  derivatives.
• Its important to be fast as time is your enemy
  on the AP Exam.
• When you think you know the answer,
• (or if you give up ) click to get to
  the next slide to see if you were correct.

3
Whats the definition of f (c)?
4
Theres 2 of them!

• You really need to know both of these thoroughly!

5
Whats the definition of f (x)?

• This is a bit differentinstead of the limit
  being a numberit will be a function related to
  f(x)
• Mrs. Cantey used to be soooo cute! ?

6
Got it??

• Notice that in the second definition, the
  arrow points to the independent variable in the
  final answer.

7
What does the derivative, f (x) tell you about
f(x)?

• (There are many answers
• how many do you know?)

8
Heres a few

• Slope of the curve
• Instantaneous rate of change in f(x)
• Rate of change
• If f(x) is positionthen velocity
• If f(x) is velocitythen acceleration

9
Now

• Here come a series of questions
• concerning basic derivatives you should
• know very quickly!

10
Ready?
11
d/dx(sin(x)) ??

• Whats your answer?

12
cos(x)

• Were you right?
• (click for next question)

13
d/dx(mxn)

• Whats your answer?

14
mnxn-1

• Were you right?
• (click for next question)

15
d/dx(ef(x))

• What is your answer?

16
f (x) ef(x)

• Were you right?
• (click for next question)

17
d/dx(lnx)

• What is your answer?

18
1/x

• Were you right?
• (click for next question)

19
d/dx(ax)

• Whats yer answereh?

20
(ln a) ax

• Were you right?
• (click for next question)
• I dig this calculus stuff!!

21
d/dx (logax)

• Whats your answer?

22
Aghh! Never can remember this one!!!

• Do it this way logax ln x / ln a
• (ln a is just a constant)
• so the answer is 1
• 
  ---------
• x ln a
• Ready?

23
d/dx(cos x)

• Whats your answer?
• (yeah yeah yeahback to trig!)

24
- sin x

• Did you remember the - ??
• Were you right?
• (click for next question)

25
d/dx(tan x)

• Answer it baby!!

26
sec2x

• Were you right?
• (click for next question)

27
d/dx(cot x)

• The ugly step sister!

28
- csc2x

• Ya ya ya!!
• Were you right?
• (click for next question)

29
d/dx(sec x)

• I call this one the wart derivative!

30
sec x tan x

• Were you right?
• (click for next question)
• (Its the TRIG family!!)

31
d/dx(csc x)

• Get er done!!!

32
- csc x cot x

• I think theres a pattern, eh?
• (click for next question)

33
d/dx(ln f(x))

• Ready?

34
f (x) / f(x)

• Were you right?
• (click for next question)

35
d/dx (af(x))

• Go go go go!!!!!

36
(ln a) f (x) af(x)

• Were you right?
• (click for next question)
• Hit that ball!!

37
d/dx (tan-1x)

• Oh yeah! Inverse trig mama!

38
1/(1x2)

• Were you right?
• (click for next question)
• Its not raining is it?

39
d/dx(sin-1x)

• You can do this one in your sleep!
• Right?

40
Did ya get it??

• Lets keep going!!!

41
d/dx(cos-1x)

• If you knew the last one, ya know this
  one.right??

42
Oh yeah!
43
d/dx(sec-1x)

• This ones a bit harder.

44
Dah dah!!!

• Did you know that one??

45
d/dx(x)

• This one is a bit esoteric

46
If you knew it was
B PROUD of ya!
47
d/dx(f(x)g(x))

• Its the Product Rule!
• (dont flush it!!!)

48
f(x)g(x)g(x)f (x)

• YeahI know you know itbut Im afraid
• Promise me you will remember to use the product
  rule every time you take the derivative of a
  product! OK??

49
d/dx(f(x)/g(x))

• Its the merry mighty Quotient Rule!!

50
Pleeeezzze!Dont get the numerator backwards.
51
d/dx(f(g(x)))

• Chain chain chain. ?
• Chain of foooools! ? ? ?

52
Yeah yeahyoure good now

• But what about that first week of May?
• Dont forget all three steps!
• Deriv of outside
• Copy inside
• Multiply by the baby

53
d/dx(y)

• Are you confused yet?

54
dy/dx

• Did I trick you ?

55
d/dx(xx)

• Hmmmmm.

56
Ha haI did trick you!

• Variable to a variable power - you cant do
  itunless you take the natural log first
• y xx
• ln y x ln x
• (1/y)(dy/dx)x(1/x) 1lnx
• dy/dx y(1lnx)

57
OKswitching to Related Rates

• For the rest of the questions, pretend you are
  doing a related rates problem.
• That ist is the independent variable
• For each expression, give the derivative as you
  would in a Related Rates problem.
• Here we go!!!

58
y
59
dy/dt

• Ready to proceed?

60
Heres a harder one
61
Did ya get it??

• dr/dt is the babyr is the mama

62
xy

• Dont be fooled!!!!

63
Ya YaProduct Rule!!

• You promised you would not forget!
• x (dy/dt) y (dx/dt)

64
sin x

• Whoo whoo dee whoo!!

65
(cos x) (dx/dt)

• Welldid you remember there was a baby?

66
Last one!

• Dont mess up!
• y2

67
(2y)(dy/dt)

• Yeah! Thats it for derivatives!
• Run this power point again if you are not 100
  certain you have mastered derivatives!

68
Goodbye!

• Be sure to run the power points for
• Integrals 2) Miscellaneous Topics
• Pre-Calc Topics 4) Sequences and Series
• 5) Other BC/BCD Topics