Rules of Differentiating

Question 1

constant rule
f(x) = 5

Answer 1

f’(x) = 0
* for any # and constant, the derivate is 0
* ⅇ, π, c k are constants, NOT VARIABLES

Question 2

power rule
f(x) = x⁵

Answer 2

f’(x) = 5x⁴

• take the power and bring it in front of the X*
• reduce the power by 1

Question 3

power rule
f(x) = x

Answer 3

f’(x) = 1

• f’(x) = 1x⁰ = 1 × 1 = 1
• take the power and bring it in front of the X
• reduce the power by 1

Question 4

constant multiple rule
y = 4x³

Answer 4

y’ = 12x²
* coefficient has no effect
* ignore coefficient
* differentiate w/rules
* simplify by multiplying by coefficient

Question 5

constant multiple rule
y = 5x

Answer 5

y’ = 5
* coefficient has no effect
* ignore coefficient
* differentiate w/rules
* simplify by multiplying by coefficient

Question 6

constant multiple rule
y = πx

Answer 6

y’ = π
* coefficient has no effect
* ignore coefficient
* differentiate w/rules
* simplify by multiplying by coefficient
* ⅇ, π, c k are constants, NOT VARIABLES

Question 7

constant multiple rule
y = π³

Answer 7

y’ = 0
* coefficient has no effect
* ignore coefficient
* differentiate w/rules
* simplify by multiplying by coefficient
* ⅇ, π, c k are constants, NOT VARIABLES

Question 8

constant multiple rule
y = 5x + 2K³

Answer 8

y’ = (5 + 0) = 5
* coefficient has no effect
* ignore coefficient
* differentiate w/rules
* simplify by multiplying by coefficient
* ⅇ, π, c k are constants, NOT VARIABLES

Question 9

sum rule / difference rule
y = 2x⁶ + x³ + x² + x + 10

Answer 9

y’ = 12x⁵ + 3x² + 2x + 1
* find the derivative of each term separately

Question 10

constant rule
y = ⅇ

Answer 10

y’ = 0

• e, π, c k are constants, NOT VARIABLES

Question 11

natural exponential function
y = ⅇ^x

Answer 11

y’ = ⅇ^x
* derivative of ⅇ^x is itself

Question 12

natural exponential function
y = 2^x

Answer 12

y’ = 2^x ln 2
* if the base is a number other than ⅇ, then multiply the derivative by the natural log of the base
* base > 0 and b ≠ 1

Question 13

logarithmic functions
f(x) = ln x

Answer 13

f’(x) = 1/x

Question 14

logarithmic functions
f(x) = ln |x|

Answer 14

f’(x) = 1/x
* absolute bars do not allow any negatives in the answer

Question 15

product rule
f(x) = x³ × sin(x)

Answer 15

Question 16

quotient rule

Answer 16

Question 17

derivative of trig functions
y = sin x

Answer 17

y’ = cos x

Question 18

derivative of trig functions
y = cos x

Answer 18

y’ = -sin x

Question 19

derivative of trig functions
y = tan x

Answer 19

y’ = sec² x

Question 20

derivative of trig functions
y = csc x

Answer 20

y’ =-csc x cot x

Question 21

derivative of trig functions
y = sec x

Answer 21

y’ =sec x tan x

Question 22

derivative of trig functions
y = cot x

Answer 22

y’ =-csc² x

Question 23

logarithmic functions

Answer 23

• If the log base is a number other than ⅇ, divide by the natural log of the base

Question 24

logarithmic functions

Answer 24

• If the log base is a number other than ⅇ, divide by the natural log of the base
• log x = log₁₀x

Question 25

tower function
f(x) = x^x

Answer 25

Question 26

logarithmic functions
y = ln 4x

Answer 26