Derivative Rules

Question 1

Which two derivative functions are identity functions?

Answer 1

D(0), D(e^x)

Question 2

D(0)

Answer 2

0

Question 3

D(e^x)

Answer 3

e^x

Question 4

Power Rule

Answer 4

D(x^n) = nx^(n-1)

Question 5

D(c)

Answer 5

0

Question 6

D(2)

Answer 6

0

Question 7

Product Rule

Answer 7

D[f(x)g(x)] = f(x)g’(x) + g(x)f’(x)

Question 8

Quotient Rule

Answer 8

D[f(x)/g(x)] = (g(x)f’(x) - f(x)g’(x))/([g(x)]^2)

Question 9

Define e

Answer 9

lim n-inf ((1+(x/n))^n)

Question 10

How do you find intervals of increase/decrease?

Answer 10

1. Find f’
2. Find where f’=0
3. Use sign chart

Question 11

If there are no intervals, then how do you know if f’>0 or f’<0?

Answer 11

Set x to a value and solve for f’. If that value is >0, then all x values will be >0 due to the IVT.

Question 12

Equation for objects in free fall

Answer 12

h(t)=-0.5gt^2 + v_0t + s_0

h:height(positin)
t:time
V_0:Initial velocity
S_0:Initial position
g:gravitational constant

Question 13

Theorem for f’(x)=0

Answer 13

1. f’(x) = 0 when: f(x) is at a smooth rel. min/max
2. when f increases over (a,b), then f’>0 over (a,b)
3. when f decreases over (a,b), then f’<0 over (a,b)

Question 14

Limit definition using h

Answer 14

lim_(h-0) ((f(x+h) - f(x))/h)

Question 15

Limit definition using x and a

Answer 15

lim_(x-a) (f(x) - f(a))/(x-a)

Question 16

Finding impact velocity

Answer 16

1. Find time when object returns to ground
2. Use v(t), which equals h’(t)

Question 17

D(sin x)

Answer 17

cos x

Question 18

D(cos x)

Answer 18

-sin x

Question 19

D(tan x)

Answer 19

sec^2 x

Question 20

D(csc x)

Answer 20

-csc x cot x

Question 21

D(sec x)

Answer 21

sec x tan x

Question 22

D(cot x)

Answer 22

-csc^2 x

Question 23

Chain Rule

Answer 23

D(f(g(x))) = f’(g(x)) * g’(x)

Question 24

Generalized Power Rule

Answer 24

D(f(x)^p) = p * f(x)^(p-1) * f’(x)

Question 25

Reciprocal Rule

Answer 25

D(1/f(x)) = (f'(x)/f(x)^2)

Question 26

D(c*f(x))

Answer 26

c * f'(x)

Question 27

D(a^x)

Answer 27

a^x * ln(a)

Question 28

D(log_a(x))

Answer 28

1/(x ln(a))

Question 29

Equation for tangent line

Answer 29

y=f'(a)(x-a)+f(a)

Question 30

D(c*e^x)

Answer 30

c*e^x

Question 31

Logarithmic differentiation

Answer 31

f'(x)=f(x) * D(ln(f(x)))

Question 32

Generalized Log Rule for Any Base

Answer 32

f'(x)/(ln(b) * f(x))

Question 33

Power Rule for Logs

Answer 33

D(ln(f(x)^p)) = D(p*ln(f(x))) = p * D(ln(f(x)))

Question 34

Generalized Exponential Rule for Any Base

Answer 34

D(b^f(x)) = ln(b) * b^f(x) * f'(x)

Question 35

Generalized Log Rule for Base e

Answer 35

D(ln(f(x))) = f'(x)/f(x)

Question 36

Generalized Exponent Rule for Base e

Answer 36

D(e^f(x)) = e^f(x) * f'(x)

Question 37

Generalized Square Root Rule

Answer 37

D(sqrt(f(x))) = f'(x)/(2sqrt(x))

Question 38

D(sin^-1 (x))

Answer 38

1/sqrt(1-x^2)

Question 39

D(cos^-1 (x))

Answer 39

-1/sqrt(1-x^2)

Question 40

D(tan^-1 (x))

Answer 40

1/(x^2 + 1)

Question 41

D(csc^-1 (x))

Answer 41

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

Question 42

D(sec^-1 (x))

Answer 42

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

Question 43

Gravitational constant of earth (metric)

Answer 43

9.81 m/s^2

Question 44

Gravitational constant of earth (imperial)

Answer 44

32 ft/s^2

Question 45

Equation for population growth/decline

Answer 45

P(t) = p(0) * e^(k * t)

Question 46

Equation for population growth/decline (derivative)

Answer 46

(dP/dt) = k * P(t)

Question 47

What does k equal?

Answer 47

k= (dP/dt)/P(t)

Question 48

Equation for temperature

Answer 48

T(t) = (T(0)-T_s)e^(kt) + T_s

Question 49

Equation for rate of change in temperature

Answer 49

(dP/dt) = k(T(0)-T_s)

Question 50

sin (x) = 0

Answer 50

x = pi * n

Question 51

cos(x) = 0

Answer 51

x = (2n + 1)(pi/2)

Question 52

Solution to b^x = 0

Answer 52

No solution

Question 53

Equation for temp difference between surrounding and object

Answer 53

y(t) = y(0)e^(kt)

Question 54

Equation for change in temp difference between surrounding and object

Answer 54

(dy/dt) = yk