Flashcards in Derivatives Deck (78):

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## Slop of the Tangent line

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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]

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## Slope

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Rise over run

y/x

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## Derivate (∂) (d/dx)

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

∂=f'(x)= Lim h→0 (f(x+h)-f(x))/h

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## Average rate of change

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

Aka: secant line

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## Secant line

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Average rate of change between two points

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

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## Differentiable if...

### Continuous, no other criteria

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## Derivative of any constant

### Zero

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

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

nx^(n-1)

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## Derivative of a Function multiplied by a constant

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Equal to the same constant multiplied by the function derivative

∂[c*f(x)]=c*f'(x)

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## Derivative of two functions added/subtracted together

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Equal to the derivatives of those functions added/subtracted together

∂[f(x)±g(x)]=f'(x)±g'(x)

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## Derivative of e^x

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Goes unchanged

∂e^x = e^x

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## Second derivative [f"(x)]

### ∂[f'(x)]=f"(x)

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## Any-order derivative formula

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[[∂f(x)]^n]/[∂(x^n)]

Formula to the n-th derivative

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##
Derivative of the product of two functions

∂[f(x)*g(x)]

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Sum of the products with the derivative switching places

f'(x)*g(x)+f(x)*g'(x)

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##
Derivative of the quotient of two functions

∂[f(x)/g(x)]

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Difference of the products with the derivative switching places, over second function squared

[f'(x)*g(x)-f(x)*g'(x)]/[g(x)^2]

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## Derivative of e^(kx)

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Unchanged but multiplied by k

k*e^(kx)

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## Derivative sin(x)

### Cos(x)

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## Trigonometric derivative chain

### Sin(x)→Cos(x)→-Sin(x)→-Cos(x)→repeat

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## Derivative -sin(x)

### -Cos(x)

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## Derivative cos(x)

### -sin(x)

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## Derivative -cos(x)

### Sin(x)

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## Derivative tan(x)

### Sec^2 (x)

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## Derivative -tan(x)

### -sec^2 (x)

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## Derivative -cot(x)

### Csc^2 (x)

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## Derivative cot(x)

### -csc^2 (x)

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## Differentiation process

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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

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## Derivative sec(x)

### Sec(x)*tan(x)

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## Derivative -sec(x)

### -Sec(x)*tan(x)

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## Derrivative csc(x)

### -csc(x)*cot(x)

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## Derrivative -csc(x)

### csc(x)*cot(x)

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## Instantaneous values

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Lim (x-xi)→0= (f(x)-f(xi))/(x-xi)

When (x+xi)=[desiredValue]

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## Average cost

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

C(x)- cost of producing x-items

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## Marginal cost

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The approximate cost of producing one more item after youyoure first x items

C'(x)

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## Chain rule- for (f o g)

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Derivative of the first function, of interior function, times derivative of interior function of contents

f(g(x))=f'(g(x))*g'(x)

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## Derivative for a function to a power

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∂[f(x)^n]=f'(x)*n(f(x))^(n-1)

Derivative f(x) times exponent rule

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## Implicit differentiation (dy/dx)

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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

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## Derivative of ln x

### 1/x

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

### (b^x)*ln(b)

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## Derivative log-b x

### 1/[x*ln(x)]

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## Derivative of sin^-1(x)

### 1/√(1-x^2)

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## Derivative of tan^-1(x)

### 1/(1+x^2)

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## Derivative cos^-1(x)

### -1/√(1-x^2)

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## Derivative cot^-1(x)

### -1/(1+x^2)

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## Derivative sec^-1(x)

### 1/(|x|*√(x^2-1))

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## Derivative csc^-1(x)

### -1/(|x|*√(x^2-1))

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## Steps for related rate problems

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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

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## Absolute maximum

### The greatest output on an entire curve

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## Absolute Minimum

### The least output on an entire curve

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## Extreme Value Theorem

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

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## Local minimum

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

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## Local maximum

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

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## Extreme point theorem

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The derivative of a maxima or a minima is always zero

∂[extrema]=0

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## The derivative of a maxima or a minima

### Is always zero

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## Finding extrema values

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1) Solve for the derivative

2) Set equal to zero

3) Simplify

4) Isolate x

5) Use algebra until you find a set value

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## Concave up

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Positive Derivative

Curves up, approaching infinity

Visualize an upward opening parabola

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## Concave down

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Negative derivative

Curves down

Visualize a downward opening parabola

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## Inflection point

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Any point at which the derivative goes from + to -, or from - to +

Always f"(x)=0

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## Second derivative test

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When f'(x)=0

If f"(x)>0 → minimum

If f"(x)<0 → maximum

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## Objective function

### Quantity you wish to maximize

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## Maximizing objective functions

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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

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## Linear approximation

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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)

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## Differntials

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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)

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## Mean Value Theorem (Rolle's Theorem)

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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)

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## Lhopital's rule

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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

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## Growth rates

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f(x) grows faster than g(x) if

Lim→∞ g(x)/f(x)=0

Or

Lim→∞ f(x)/g(x)=∞

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## The rates of growth (dy/dt)

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Growth Rates=∂[P*e^(r*t)]=P*r*e^(r*t)=r*A(t)

Described as dy/dt

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## Rate constant (k)

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The rate by which P*e^(r*t) grows exponentially

Here, it is the 'r'

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## Relative growth rate

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Rate divided by current output

(dy/dt)/y

Always equal to 'k' (or 'r')

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## Doubling time

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Time it takes to before the initial value doubles

T2=ln2/k

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## Exponential decay

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Describes how P decreases with time

Takes the form: P*-e^(r*t)

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## Halflife

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Time it takes for the decay function to reach half its original value

T(1/2)=ln(2)/k

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## Economic Elasticity

### D

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## Atomic Kinetics

### S

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## Newton's Methods

### S

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## Oscilators

### S

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## Partial derivatives

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1) Pick the variable indicated in the problem

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

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## Newtons Notation/Lagrange's Notation

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Marks derivatives as

F'(x)

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