Derivatives Flashcards Preview

β–Ί Calculus > Derivatives > Flashcards

Flashcards in Derivatives Deck (78):
1

Slope

Rise over run
y/x

2

Derivate (βˆ‚) (d/dx)

Slope of the tangent line as it changes with (x)
βˆ‚=f'(x)= Lim hβ†’0 (f(x+h)-f(x))/h

3

Average rate of change

m=(f(x)-f(a))/(x-a)

Aka: secant line

4

Secant line

Average rate of change between two points
m=(f(x)-f(a))/(x-a)

5

Differentiable if...

Continuous, no other criteria

6

Derivative of any constant

Zero

7

Derivative of a variable to a power (x^n)

Exponent times base to the power of (exponent-1)
nx^(n-1)

8

Derivative of a Function multiplied by a constant

Equal to the same constant multiplied by the function derivative
βˆ‚[c*f(x)]=c*f'(x)

9

Derivative of two functions added/subtracted together

Equal to the derivatives of those functions added/subtracted together
βˆ‚[f(x)Β±g(x)]=f'(x)Β±g'(x)

10

Derivative of e^x

Goes unchanged

βˆ‚e^x = e^x

11

Second derivative [f"(x)]

βˆ‚[f'(x)]=f"(x)

12

Any-order derivative formula

[[βˆ‚f(x)]^n]/[βˆ‚(x^n)]

Formula to the n-th derivative

13

Derivative of the product of two functions
βˆ‚[f(x)*g(x)]

Sum of the products with the derivative switching places
f'(x)*g(x)+f(x)*g'(x)

14

Derivative of the quotient of two functions
βˆ‚[f(x)/g(x)]

Difference of the products with the derivative switching places, over second function squared
[f'(x)*g(x)-f(x)*g'(x)]/[g(x)^2]

15

Derivative of e^(kx)

Unchanged but multiplied by k
k*e^(kx)

16

Derivative sin(x)

Cos(x)

17

Trigonometric derivative chain

Sin(x)β†’Cos(x)β†’-Sin(x)β†’-Cos(x)β†’repeat

18

Derivative -sin(x)

-Cos(x)

19

Derivative cos(x)

-sin(x)

20

Derivative -cos(x)

Sin(x)

21

Derivative tan(x)

Sec^2 (x)

22

Derivative -tan(x)

-sec^2 (x)

23

Derivative -cot(x)

Csc^2 (x)

24

Derivative cot(x)

-csc^2 (x)

25

Differentiation process

1) chain rule first, always
2) turn exponent to nx^(n-1)
3) factor out the constant and e^x multipliers
4) use product rule, use quotient rule
5) use sum/difference rule
6) replace trig functions with their variables

26

Derivative sec(x)

Sec(x)*tan(x)

27

Derivative -sec(x)

-Sec(x)*tan(x)

28

Derrivative csc(x)

-csc(x)*cot(x)

29

Derrivative -csc(x)

csc(x)*cot(x)

30

Instantaneous values

Lim (x-xi)β†’0= (f(x)-f(xi))/(x-xi)
When (x+xi)=[desiredValue]

31

Average cost

(C(x)-C(xi))/(x-xi)

C(x)- cost of producing x-items

32

Marginal cost

The approximate cost of producing one more item after youyoure first x items
C'(x)

33

Chain rule- for (f o g)

Derivative of the first function, of interior function, times derivative of interior function of contents
f(g(x))=f'(g(x))*g'(x)

34

Derivative for a function to a power

βˆ‚[f(x)^n]=f'(x)*n(f(x))^(n-1)
Derivative f(x) times exponent rule

35

Implicit differentiation (dy/dx)

1) Define y as y(x)
2) Take the derivative of each term
3) Rearrange to from y(x)*y'(x)
3) Turn y(x)*y'(x) into y*dy/dx
4) Isolate the term containing y*dy/dx
5) Reduce to its simplest form

36

Derivative of ln x

1/x

37

Derivative of a constant raised to a variable (b^x)

(b^x)*ln(b)

38

Derivative log-b x

1/[x*ln(x)]

39

Derivative of sin^-1(x)

1/√(1-x^2)

40

Derivative of tan^-1(x)

1/(1+x^2)

41

Derivative cos^-1(x)

-1/√(1-x^2)

42

Derivative cot^-1(x)

-1/(1+x^2)

43

Derivative sec^-1(x)

1/(|x|*√(x^2-1))

44

Derivative csc^-1(x)

-1/(|x|*√(x^2-1))

45

Steps for related rate problems

1) Write equations that express basic relationships between variables
2) Introduce rates of change by differentiating the appropriate equations with respect to time
3) Introduce rates of change by differentiating the approprite equations with respect to time
4) Substitute known values and solve for the desired quantity
5) check that the untis are reasonable

46

Absolute maximum

The greatest output on an entire curve

47

Absolute Minimum

The least output on an entire curve

48

Extreme Value Theorem

On a closed interval [a,b], the curve has both a minimum and a maximum value

49

Local minimum

The least possible output on the interval [a,b]

50

Local maximum

The greatest possible output on the interval [a,b]

51

Extreme point theorem

The derivative of a maxima or a minima is always zero

βˆ‚[extrema]=0

52

The derivative of a maxima or a minima

Is always zero

53

Finding extrema values

1) Solve for the derivative
2) Set equal to zero
3) Simplify
4) Isolate x
5) Use algebra until you find a set value

54

Concave up

Positive Derivative
Curves up, approaching infinity
Visualize an upward opening parabola

55

Concave down

Negative derivative
Curves down
Visualize a downward opening parabola

56

Inflection point

Any point at which the derivative goes from + to -, or from - to +
Always f"(x)=0

57

Second derivative test

When f'(x)=0

If f"(x)>0 β†’ minimum
If f"(x)

58

Objective function

Quantity you wish to maximize

59

Maximizing objective functions

1) Write the functions that you know
2) Eliminate all but one of the independent variables via substitution
3) Use algebra to convert this to an algebraic function
4) Calculate the derivative
5) set equal to zero
6) Solve for x

60

Linear approximation

Use the output of the line tangent to a nearby point to approximate the actual function output at that value, f(a) is the tangent line

f(x)β‰ˆf(a)+f'(x)(x-a)

61

Differntials

Functions that describe variations between the line tangent to a nearby point and the actual function output at that value, f(a) is the tangent line

Ξ”y=f(a+Ξ”x)-f(a)

62

Mean Value Theorem (Rolle's Theorem)

On every interval [a,b] there is a value 'c' between 'a' and 'b', equal to the average slope on that interval

f'(c)=[f(b)-f(a)]/(b-a)

63

Lhopital's rule

Any limit f(x)/g(x)= same limit f'(x)/g'(x)

True if f(x) and g(x) limits are 0/0
∞/∞
0*∞
∞-∞
1^∞
0^0
∞^0

64

Growth rates

f(x) grows faster than g(x) if
Limβ†’βˆž g(x)/f(x)=0
Or
Limβ†’βˆž f(x)/g(x)=∞

65

The rates of growth (dy/dt)

Growth Rates=βˆ‚[P*e^(r*t)]=P*r*e^(r*t)=r*A(t)
Described as dy/dt

66

Rate constant (k)

The rate by which P*e^(r*t) grows exponentially
Here, it is the 'r'

67

Relative growth rate

Rate divided by current output
(dy/dt)/y
Always equal to 'k' (or 'r')

68

Doubling time

Time it takes to before the initial value doubles
T2=ln2/k

69

Exponential decay

Describes how P decreases with time
Takes the form: P*-e^(r*t)

70

Halflife

Time it takes for the decay function to reach half its original value
T(1/2)=ln(2)/k

71

Economic Elasticity

D

72

Atomic Kinetics

S

73

Newton's Methods

S

74

Oscilators

S

75

Partial derivatives

1) Pick the variable indicated in the problem
2) calculate the derivative as if all the other variables were actually constants

76

Newtons Notation/Lagrange's Notation

Marks derivatives as
F'(x)

77

Leibniz's Notation

Mark derivarves as
d/dx

78

Slop of the Tangent line

Instantaneous Rate of change for the curve,
Slope at a point
Lim (x-xi)β†’0 for (f(x)-f(xi))/(x-xi)
So long as x=[point in question]