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Practice > Basic Calculus > Flashcards

Basic Calculus Flashcards

1
Q

d/dx (sin 5x)

A

5 cos 5x

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2
Q

d/dx (tan 9x)

A

9 sec^2 (9x)

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3
Q

d/dx (cot 7x)

A

-7 csc^2 (7x)

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4
Q

d/dx (csc x^2)

A

-2x csc(x^2) cot(x^2)

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5
Q

Quotient Rule

A

lo d’hi - hi d’lo/ (lo)^2

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6
Q

Product Rule

A

1st d’2nd + 2nd d’1st

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7
Q

d/dx (cos u)

A

-sin u du/dx

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8
Q

d/dx (sin u)

A

cos u du/dx

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9
Q

d/dx (tan u)

A

sec^2 u du/dx

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10
Q

d/dx (cot u)

A

-csc^2 u du/dx

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11
Q

d/dx (sec u)

A

sec u tan u du/dx

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12
Q

If f(x) = c where c is a constant
then f’(x) = 0

A

Constant Rule

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13
Q

If f(x) = x, then f’(x) = 1

A

Identity Rule

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14
Q

If f(x) = x^n, then f’(x) = nx^n-1

A

Power Rule

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15
Q

If f(x) = Cx^n, then f’(x) = Cnx^n-1

A

Constant Multiple Rule

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16
Q

If f(x) = u(x) +/- v(x)
then f’(x) = u’(x) +/- v’(x)

A

Sum and Difference Rule

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17
Q

If f(x) = u(x) • v(x)
then f’(x) = u(x) - v’(x) + v(x) - u’(x)

A

Product Rule

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18
Q

If f(x) = u(x)/ v(x)
then f’(x) = v(x) - u’(x) - u(v) • v’(x)
/ [u(x)]^2

A

Quotient Rule

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19
Q

If f(x) = [u(x)]^n
then f’(x) = n [u(x)] ^ n-1 [u’(x)]

A

Chain Rule

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20
Q

e^5x

A

5e^5x

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21
Q

d/dx (e^2x)

A

2e^2x

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22
Q

d/dx (e^cos 3x)

A

u = cos 3x

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23
Q

cos (5x^2)

A

u = 5x^2
du = 10x

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24
Q

sin (3x^2)

A

u = 3x^2
du = 6x

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25
Q

tan (5x)

A

u = 5x
du = 5

26
Q

tan ( x+3 )

A

u = x+3
du = 1

27
Q

tan ( 2x^3 )

A

u = 2x^3
du = 6x^2

28
Q

u = cos 3x

du = ?

A

du = -3 sin 3x

29
Q

d/dx ( xe^2x )

x • 2e^2x + e^2x • 1

ANS: ?

A

ANS: 2xe^2x + e^2x

30
Q

u = 2x^2 + x

du = ?

A

du = 4x + 1

31
Q

u = x^2

du = ?

A

du = 2x

32
Q

u(x) = sin e^3x

u’(x) = ?

A

u’(x) = (cos e^3x) (3e^3x)

33
Q

u(x) = sin (3x^2)

u’(x) = ?

A

u’(x) = 6x cos (3x^2)

34
Q

u(x) = tan (5x)

u’(x) = ?

A

u’(x) = 5 sec^2 (5x)

35
Q

u(x) = tan (x+3)

u’(x) = ?

A

u’(x) = 1 sec^2 (x+3) or sec^2(x+3)

36
Q

u(x) = sin (2x)

u’(x) = ?

A

u’(x) = 2 cos (2x)

37
Q

u(x) = sin ( 4x )

u’(x) = ?

A

u’(x) = 4 cos (4x)

38
Q

u = sin 3x

u’ = ?

A

u’ = 3 cos 3x

39
Q

Product Rule: 1st • d’2nd + 2nd • d’1st

u(x) = 4x^2

u’(x) = 8x

v(x) = tan (2x^3)

v’(x) = 6x^2 sec^2 (2x^3)

ANS: ?

A

ANS: 4x^2
( 6x^2 sec^2 (2x^3) )
+ tan (2x^3) • 8x

40
Q

Product Rule: 1st • d’2nd + 2nd • d’1st

u(x) = sin (2x)
u’(x) = 2 cos (2x)

v(x) = cos (3x)
v’(x) = -3 sin (3x)

ANS: ?

A

ANS: sin (2x) [ -3 sin (3x) ]
+ cos (3x) [ 2 cos (2x) ]

41
Q

e^u • du

e^cos3x • -3 sin 3x

ans: ?

A

ans: -3e^cos3x sin 3x
or -3 sin 3x (e^cos3x)

42
Q

a^u ln a • du

Dx (10^x) = ?

A

10^x ln 10 • 1

a = 10
u = x
du = 1

43
Q

Quotient Rule: lo d’hi - hi d’lo / (lo)^2

d/dx ( x/ 10^x + 8 )

ANS: ?

Dx(10^x) = 10^x ln 10

A

ANS: ( 10^x + 8 ) (1) - x ( 10^x ln 10 )
/ ( 10^x + 8 ) ^2

44
Q

Dx ( ln u ) = ?

A

u’ / u

45
Q

ln u^2

A

2 (ln u)

46
Q

log 10 x^2

A

2 ( log 10 x )

47
Q

x^2 ( ln x^5 )

A

x^2 ( 5 ln x )

48
Q

ln ( x - 5 ) ^1/2 - ln ( x^2-7 )

A

1/2 ln ( x-5 ) - ln ( x^2 - 7 )

49
Q

d/dx (a^u) = ?

A

a^u ln a du/dx

50
Q

sin x

A

1 / csc x

51
Q

csc x

A

1 / sin x

52
Q

cos x

A

1 / sec x

53
Q

sec x

A

1 / cos x

54
Q

tan x

A

1 / cot x

55
Q

cot x

A

1 / tan x

56
Q

tan x

A

sin x / cos x

57
Q

cot x

A

cos x / sin x

58
Q

? = 1

A

sin^2 x + cos^2 x

59
Q

? = sec^2 x

A

tan^2 x + 1

60
Q

? = csc^2 x

A

1 + cot^2 x

Practice (4 decks)

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