derivatives

Question 1

Constant

Answer 1

Function: C
Derivative: 0

Question 2

Line

Answer 2

Function: x Derivative: 1
Function: ax Derivative: a

Question 3

Square

Answer 3

Function: x^2
Derivative: 2x

Question 4

Square Root

Answer 4

Function:√x
Derivative: (½)x-½

Question 5

Exponential

Answer 5

Function: e^x Derivative: e^x
Function: a^x Derivative: ln (a) a^x

Question 6

Logarithms

Answer 6

Function: ln(x) Derivative: 1/x
Function: loga(x) Derivative: 1 / (x ln(a))

Question 7

TRIG sin(x)

Answer 7

Function: sin(x)
Derivative: cos(x)

Question 8

TRIG cos(x)

Answer 8

Function: cos(x)
Derivative: -sin(x)

Question 9

TRIG tan(x)

Answer 9

Function: tan(x)
Derivative: sec^2 (x)

Question 10

Rules: Multiplication by constant

Answer 10

F: cf D: cf’

Question 11

Power Rule

Answer 11

F:x^n
D: nx^n−1

Question 12

Sum Rule

Answer 12

F: f + g
D: f’ + g’

Question 13

Difference Rule

Answer 13

F: f - g
D: f’ − g’

Question 14

Product Rule

Answer 14

F: fg
D: f g’ + f’g

Question 15

Quotient Rule

Answer 15

F: f/g
D: (f’ g − g’ f )/g^2

Question 16

Reciprocal Rule

Answer 16

F: 1/f
D: −f’/f^2

Question 17

Chain Rule (as composition of functions)

Answer 17

F: f º g
D: (f’ º g) × g’

Question 18

Chain Rule (using ‘)

Answer 18

F: f(g(x))
D: f’(g(x))g’(x)

Question 19

Chain Rule ((using d/dx )

Answer 19

F: dy/dx = dy/du * du/dx

which just means derivative of the whole x derivative of the inside

Question 20

d/dx

Answer 20

“The derivative of” is also written d/ dx

Question 21

Exponentiation: Fractional Exponents

Answer 21

x^m/n = the n root of x^m or (n root of x)^m

Question 22

Exponentiation: Power of a product

Answer 22

(xy)^a=x^ay^a

Question 23

natural log Product Rule

Answer 23

ln(xy)=ln(x)+ln(y)

Question 24

natural log Quotient Rule

Answer 24

ln(x/y)=ln(x)−ln(y)

Question 25

Log of e

Answer 25

ln(e)=1

Question 26

Log of one

Answer 26

ln(1)=0

Question 27

Log reciprocal

Answer 27

ln(1/x)=−ln(x)