Calculus Integration and Derivatives Rules

Question 1

What is the constant rule
d/dx [c] = ?

Answer 1

Derivative of a constant is equal to zero.

Question 2

Power Rule for Derivatives

Answer 2

d/dx [x^n] = nx^(n-1)

Question 3

Product Rule for Derivative
d/dx [f(x)g(x)] = ?

Answer 3

f ‘ (x) g(x) + f(x) g’ (x)

Question 4

Quotient Rule for Derivatives
d/dx [f(x)/g(x)] = ?

Answer 4

[g(x)f ‘(x) - f(x) g’ (x)] / [g(x)]^2

Question 5

Exponential Rule for Derivative
d/dx b^x

Answer 5

b^x * d/dx (x)

Question 6

Logarithmic Rule for Derivatives
d/dx ln (x)

Answer 6

1/x

Question 7

Constant Multiple Rule for Derivatives
d/dx [c(f(x)] = ?

Answer 7

c f ‘ (x)

Question 8

Sum Rule for Derivatives
d/dx [f(x) + g(x)] = ?

Answer 8

f ‘ (x) + g’ (x)

Question 9

Difference Rule for Derivatives
d/dx [f(x) - g(x) ] = ?

Answer 9

f ‘ (x) - g’ (x)

Question 10

Chain Rule for Derivatives
d/dx f(g(x)) = ?

Answer 10

f ‘ (g(x)) g’ (x)

Question 11

d/dx (x) = ?

Answer 11

1

Question 12

d/dx (sin x) = ?

Answer 12

cos x

Question 13

d/dx (cos x) = ?

Answer 13

-sin x

Question 14

d/dx (tan x) = ?

Answer 14

sec^2 (x)

Question 15

d/dx (sec x) = ?

Answer 15

sec x tan x

Question 16

d/dx (csc x) = ?

Answer 16

• csc x cot x

Question 17

d/dx (cot x) = ?

Answer 17

-csc^2 (x)

Question 18

d/dx (sin^-1 (x)) = ?

Answer 18

1/ sqrt (1-(x^2))

Question 19

d/dx (cos^-1 (x)) = ?

Answer 19

-1/ sqrt (1-(x^2))

Question 20

d/dx (tan^-1 (x)) = ?

Answer 20

1/sqrt (1 + x^2))

Question 21

d/dx (a^x) = ?

Answer 21

a^x (ln a)

Question 22

d/dx (e^x) = ?

Answer 22

e^x

Question 23

d/dx log base a (x) = ?

Answer 23

1/x ln a

Question 24

csc (tetha) = ?

Answer 24

1/sin (theta)

Question 25

sec (tetha) = ?

Answer 25

1/ cos (tetha)

Question 26

cot (tetha) = ?

Answer 26

1/ tan (theta)

Question 27

Integration of a dx

Answer 27

ax + C

Question 28

Power Rule for Integration

Answer 28

integration of x^n dx = [x^(n+1)]/n+1 + c, n not equal to -1.

Question 29

Integration x^-1 dx

Answer 29

ln |x|+ c

Question 30

Integration of e^x dx

Answer 30

e^x + c

Question 31

Integration a^x dx

Answer 31

a^x/ln a + c

Question 32

Integration kf(x) dx = ?

Answer 32

k integration f(x) dx.

Question 33

Integration f(x) + g(x) dx = ?

Answer 33

Integration f(x) dx + Integration g(x) dx

Question 34

Integration f(x) - g(x) dx = ?

Answer 34

Integration f(x) dx - Integration g(x) dx

Question 35

Integration sin x dx

Answer 35

-cos x + c

Question 36

Integration cos x dx

Answer 36

sin x + c

Question 37

Integration sec^2 x dx = ?

Answer 37

tan x + c

Question 38

Integration csc^2 x dx = ?

Answer 38

-cot x + c

Question 39

Integration sec x tan x dx = ?

Answer 39

sec x + c

Question 40

Integration csc x cot x dx = ?

Answer 40

-csc x + c

Question 41

sin^2 x + cos^2 x = ?

Answer 41

1

Question 42

sin^2 x = ?

Answer 42

1/2 [1- cos 2x]

Question 43

cos^2 x = ?

Answer 43

1 - sin^2 x

Question 44

sec^2 x = ?

Answer 44

1 + tan^2 x

Question 45

csc^2 x = ?

Answer 45

1 + cot^2 x

Question 46

cos^2 x = ?

Answer 46

1/2 [1 + cos 2x]

Question 47

sin (2x) = ?

Answer 47

2 sin x cos x