Derivatives

Question 1

Slope

Answer 1

Rise over run

y/x

Question 2

Derivate (∂) (d/dx)

Answer 2

Slope of the tangent line as it changes with (x)

∂=f’(x)= Lim h→0 (f(x+h)-f(x))/h

Question 3

Average rate of change

Answer 3

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

Aka: secant line

Question 4

Secant line

Answer 4

Average rate of change between two points

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

Question 5

Differentiable if…

Answer 5

Continuous, no other criteria

Question 6

Derivative of any constant

Answer 6

Zero

Question 7

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

Answer 7

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

nx^(n-1)

Question 8

Derivative of a Function multiplied by a constant

Answer 8

Equal to the same constant multiplied by the function derivative
∂[cf(x)]=cf’(x)

Question 9

Derivative of two functions added/subtracted together

Answer 9

Equal to the derivatives of those functions added/subtracted together
∂[f(x)±g(x)]=f’(x)±g’(x)

Question 10

Derivative of e^x

Answer 10

Goes unchanged

∂e^x = e^x

Question 11

Second derivative [f”(x)]

Answer 11

∂[f’(x)]=f”(x)

Question 12

Any-order derivative formula

Answer 12

[[∂f(x)]^n]/[∂(x^n)]

Formula to the n-th derivative

Question 13

Derivative of the product of two functions

∂[f(x)*g(x)]

Answer 13

Sum of the products with the derivative switching places

f’(x)g(x)+f(x)g’(x)

Question 14

Derivative of the quotient of two functions

∂[f(x)/g(x)]

Answer 14

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

Question 15

Derivative of e^(kx)

Answer 15

Unchanged but multiplied by k

k*e^(kx)

Question 16

Derivative sin(x)

Answer 16

Cos(x)

Question 17

Trigonometric derivative chain

Answer 17

Sin(x)→Cos(x)→-Sin(x)→-Cos(x)→repeat

Question 18

Derivative -sin(x)

Answer 18

-Cos(x)

Question 19

Derivative cos(x)

Answer 19

-sin(x)

Question 20

Derivative -cos(x)

Answer 20

Sin(x)

Question 21

Derivative tan(x)

Answer 21

Sec^2 (x)

Question 22

Derivative -tan(x)

Answer 22

-sec^2 (x)

Question 23

Derivative -cot(x)

Answer 23

Csc^2 (x)

Question 24

Derivative cot(x)

Answer 24

-csc^2 (x)

Question 25

Differentiation process

Answer 25

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

Question 26

Derivative sec(x)

Answer 26

Sec(x)*tan(x)

Question 27

Derivative -sec(x)

Answer 27

-Sec(x)*tan(x)

Question 28

Derrivative csc(x)

Answer 28

-csc(x)*cot(x)

Question 29

Derrivative -csc(x)

Answer 29

csc(x)*cot(x)

Question 30

Instantaneous values

Answer 30

Lim (x-xi)→0= (f(x)-f(xi))/(x-xi)

When (x+xi)=[desiredValue]

Question 31

Average cost

Answer 31

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

C(x)- cost of producing x-items

Question 32

Marginal cost

Answer 32

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

Question 33

Chain rule- for (f o g)

Answer 33

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

Question 34

Derivative for a function to a power

Answer 34

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

Question 35

Implicit differentiation (dy/dx)

Answer 35

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 ydy/dx
4) Isolate the term containing ydy/dx
5) Reduce to its simplest form

Question 36

Derivative of ln x

Answer 36

1/x

Question 37

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

Answer 37

(b^x)*ln(b)

Question 38

Derivative log-b x

Answer 38

1/[x*ln(x)]

Question 39

Derivative of sin^-1(x)

Answer 39

1/√(1-x^2)

Question 40

Derivative of tan^-1(x)

Answer 40

1/(1+x^2)

Question 41

Derivative cos^-1(x)

Answer 41

-1/√(1-x^2)

Question 42

Derivative cot^-1(x)

Answer 42

-1/(1+x^2)

Question 43

Derivative sec^-1(x)

Answer 43

1/(|x|*√(x^2-1))

Question 44

Derivative csc^-1(x)

Answer 44

-1/(|x|*√(x^2-1))

Question 45

Steps for related rate problems

Answer 45

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

Question 46

Absolute maximum

Answer 46

The greatest output on an entire curve

Question 47

Absolute Minimum

Answer 47

The least output on an entire curve

Question 48

Extreme Value Theorem

Answer 48

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

Question 49

Local minimum

Answer 49

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

Question 50

Local maximum

Answer 50

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

Question 51

Extreme point theorem

Answer 51

The derivative of a maxima or a minima is always zero

∂[extrema]=0

Question 52

The derivative of a maxima or a minima

Answer 52

Is always zero

Question 53

Finding extrema values

Answer 53

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

Question 54

Concave up

Answer 54

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

Question 55

Concave down

Answer 55

Negative derivative
Curves down
Visualize a downward opening parabola

Question 56

Inflection point

Answer 56

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

Question 57

Second derivative test

Answer 57

When f’(x)=0

If f”(x)>0 → minimum
If f”(x)

Question 58

Objective function

Answer 58

Quantity you wish to maximize

Question 59

Maximizing objective functions

Answer 59

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

Question 60

Linear approximation

Answer 60

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)≈f(a)+f’(x)(x-a)

Question 61

Differntials

Answer 61

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

Δy=f(a+Δx)-f(a)

Question 62

Mean Value Theorem (Rolle’s Theorem)

Answer 62

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)

Question 63

Lhopital’s rule

Answer 63

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

Question 64

Growth rates

Answer 64

f(x) grows faster than g(x) if
Lim→∞ g(x)/f(x)=0
Or
Lim→∞ f(x)/g(x)=∞

Question 65

The rates of growth (dy/dt)

Answer 65

Growth Rates=∂[Pe^(rt)]=Pre^(rt)=rA(t)

Described as dy/dt

Question 66

Rate constant (k)

Answer 66

The rate by which Pe^(rt) grows exponentially

Here, it is the ‘r’

Question 67

Relative growth rate

Answer 67

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

Question 68

Doubling time

Answer 68

Time it takes to before the initial value doubles

T2=ln2/k

Question 69

Exponential decay

Answer 69

Describes how P decreases with time

Takes the form: P-e^(rt)

Question 70

Halflife

Answer 70

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

Question 71

Economic Elasticity

Answer 71

D

Question 72

Atomic Kinetics

Answer 72

S

Question 73

Newton’s Methods

Answer 73

S

Question 74

Oscilators

Answer 74

S

Question 75

Partial derivatives

Answer 75

1) Pick the variable indicated in the problem

2) calculate the derivative as if all the other variables were actually constants

Question 76

Newtons Notation/Lagrange’s Notation

Answer 76

Marks derivatives as

F’(x)

Question 77

Leibniz’s Notation

Answer 77

Mark derivarves as

d/dx

Question 78

Slop of the Tangent line

Answer 78

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