Derivatives and Rules

Question 1

d/dx (Sin x)

Answer 1

Cos x

Question 2

d/dx (Cos x)

Answer 2

• Sin x

Question 3

d/dx (Tan x)

Answer 3

Sec ² x

Question 4

d/dx (Sec x)

Answer 4

Sec x Tan x

Question 5

d/dx (Csc x)

Answer 5

• Csc x Cot x

Question 6

d/dx (Cot x)

Answer 6

• Csc ² x

Question 7

d/dx (e^x)

Answer 7

e^x

Question 8

d/dx (a^x)

Answer 8

a^x * ln a

Question 9

Integral (a^x) dx

Answer 9

((a^x / ln a)+ C)

Question 10

d/dx (xⁿ)

Answer 10

n * x^n-1

Question 11

d/dx (Sin ^-1 (x / a))

Answer 11

1 / sq rt ( a² - x²)

Question 12

d/dx (Cos ^-1 (x / a))

Answer 12

-1 / sq rt (a² - x²)

Question 13

d/dx (Sec ^-1 (x / a))

Answer 13

a / (|x| * sq rt (x² - a²))

Question 14

d/dx (Csc ^-1 (x / a))

Answer 14

-a / (|x| * sq rt (x² - a²))

Question 15

d/dx (Tan ^-1 (x / a))

Answer 15

a / (x² + a²)

Question 16

d/dx (Cot ^-1 (x / a))

Answer 16

-a / (x² + a²)

Question 17

d/dx [f(x) * g(x)]

Answer 17

f’(x) * g(x) + g’(x) * f(x)

Question 18

d/dx [f(x) / g(x)]

Answer 18

( ( f’(x) * g(x) ) - ( g’(x) * f(x)) ) / g(x)²

Low * D high - high * D low all over low squared

Question 19

d/dx [f( g(x) )]

Answer 19

f’( g(x) ) * g’(x)

Question 20

d/dx ( ln x)

Answer 20

1 / x

1 / something * chain rule of something

Question 21

ln a * b

Answer 21

ln a + ln b

Question 22

ln a / b

Answer 22

ln a - ln b

Question 23

ln b^a

Answer 23

a * ln b

Question 24

log _b a

Answer 24

ln a / ln b

Question 25

Average value of f(x) from [a, b]

Answer 25

( ∫^b_a f(x) ) / ( b - a )