Basic Calculus

Question 1

d/dx (sin 5x)

Answer 1

5 cos 5x

Question 2

d/dx (tan 9x)

Answer 2

9 sec^2 (9x)

Question 3

d/dx (cot 7x)

Answer 3

-7 csc^2 (7x)

Question 4

d/dx (csc x^2)

Answer 4

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

Question 5

Quotient Rule

Answer 5

lo d’hi - hi d’lo/ (lo)^2

Question 6

Product Rule

Answer 6

1st d’2nd + 2nd d’1st

Question 7

d/dx (cos u)

Answer 7

-sin u du/dx

Question 8

d/dx (sin u)

Answer 8

cos u du/dx

Question 9

d/dx (tan u)

Answer 9

sec^2 u du/dx

Question 10

d/dx (cot u)

Answer 10

-csc^2 u du/dx

Question 11

d/dx (sec u)

Answer 11

sec u tan u du/dx

Question 12

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

Answer 12

Constant Rule

Question 13

If f(x) = x, then f’(x) = 1

Answer 13

Identity Rule

Question 14

If f(x) = x^n, then f’(x) = nx^n-1

Answer 14

Power Rule

Question 15

If f(x) = Cx^n, then f’(x) = Cnx^n-1

Answer 15

Constant Multiple Rule

Question 16

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

Answer 16

Sum and Difference Rule

Question 17

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

Answer 17

Product Rule

Question 18

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

Answer 18

Quotient Rule

Question 19

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

Answer 19

Chain Rule

Question 20

e^5x

Answer 20

5e^5x

Question 21

d/dx (e^2x)

Answer 21

2e^2x

Question 22

d/dx (e^cos 3x)

Answer 22

u = cos 3x

Question 23

cos (5x^2)

Answer 23

u = 5x^2
du = 10x

Question 24

sin (3x^2)

Answer 24

u = 3x^2
du = 6x

Question 25

tan (5x)

Answer 25

u = 5x du = 5

Question 26

tan ( x+3 )

Answer 26

u = x+3 du = 1

Question 27

tan ( 2x^3 )

Answer 27

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

Question 28

u = cos 3x du = ?

Answer 28

du = -3 sin 3x

Question 29

d/dx ( xe^2x ) x • 2e^2x + e^2x • 1 ANS: ?

Answer 29

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

Question 30

u = 2x^2 + x du = ?

Answer 30

du = 4x + 1

Question 31

u = x^2 du = ?

Answer 31

du = 2x

Question 32

u(x) = sin e^3x u'(x) = ?

Answer 32

u'(x) = (cos e^3x) (3e^3x)

Question 33

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

Answer 33

u'(x) = 6x cos (3x^2)

Question 34

u(x) = tan (5x) u'(x) = ?

Answer 34

u'(x) = 5 sec^2 (5x)

Question 35

u(x) = tan (x+3) u'(x) = ?

Answer 35

u'(x) = 1 sec^2 (x+3) or sec^2(x+3)

Question 36

u(x) = sin (2x) u'(x) = ?

Answer 36

u'(x) = 2 cos (2x)

Question 37

u(x) = sin ( 4x ) u'(x) = ?

Answer 37

u'(x) = 4 cos (4x)

Question 38

u = sin 3x u' = ?

Answer 38

u' = 3 cos 3x

Question 39

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: ?

Answer 39

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

Question 40

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: ?

Answer 40

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

Question 41

e^u • du e^cos3x • -3 sin 3x ans: ?

Answer 41

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

Question 42

a^u ln a • du Dx (10^x) = ?

Answer 42

10^x ln 10 • 1 a = 10 u = x du = 1

Question 43

Quotient Rule: lo d'hi - hi d'lo / (lo)^2 d/dx ( x/ 10^x + 8 ) ANS: ? Dx(10^x) = 10^x ln 10

Answer 43

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

Question 44

Dx ( ln u ) = ?

Answer 44

u' / u

Question 45

ln u^2

Answer 45

2 (ln u)

Question 46

log 10 x^2

Answer 46

2 ( log 10 x )

Question 47

x^2 ( ln x^5 )

Answer 47

x^2 ( 5 ln x )

Question 48

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

Answer 48

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

Question 49

d/dx (a^u) = ?

Answer 49

a^u ln a du/dx

Question 50

sin x

Answer 50

1 / csc x

Question 51

csc x

Answer 51

1 / sin x

Question 52

cos x

Answer 52

1 / sec x

Question 53

sec x

Answer 53

1 / cos x

Question 54

tan x

Answer 54

1 / cot x

Question 55

cot x

Answer 55

1 / tan x

Question 56

tan x

Answer 56

sin x / cos x

Question 57

cot x

Answer 57

cos x / sin x

Question 58

? = 1

Answer 58

sin^2 x + cos^2 x

Question 59

? = sec^2 x

Answer 59

tan^2 x + 1

Question 60

? = csc^2 x

Answer 60

1 + cot^2 x