Calculus/Derivatives

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Derivative of a function at a number [edit | edit source]

Notation[edit | edit source]

We denote the derivative of a function at a number as .

Definition[edit | edit source]

The derivative of a function at a number a is given by the following limit (if it exists):


An analagous equation can be defined by letting . Then , which shows that when approaches , approaches :


Interpretations[edit | edit source]

As the slope of a tangent line[edit | edit source]

Given a function , the derivative can be understood as the slope of the tangent line to at :

Graph of parabola and tangent line.png
Example[edit | edit source]

Find the equation of the tangent line to at .

Solution[edit | edit source]

To find the slope of the tangent, we let and use our first definition:


It can be seen that as approaches , we are left with . If we plug in for :


So our preliminary equation for the tangent line is . By plugging in our tangent point to find , we can arrive at our final equation:


So our final equation is .

As a rate of change[edit | edit source]

The derivative of a function at a number can be understood as the instantaneous rate of change of when .

The derivative as a function[edit | edit source]

So far we have only examined the derivative of a function at a certain number . If we move from the constant to the variable , we can calculate the derivative of the function as a whole, and come up with an equation that represents the derivative of the function at any arbitrary value. For clarification, the derivative of at is a number, whereas the derivative of is a function.

Notation[edit | edit source]

Likewise to the derivative of at , the derivative of the function is denoted .

Definition[edit | edit source]

The derivative of the function is defined by the following limit:

Also,

or


[edit | edit source]

Consider the sequences:

many terms containing

many terms containing

many terms containing

Therefore:

many terms containing

read as "derivative with respect to of to the power ."

Later it will be shown that this is valid for all real , positive or negative, integer or fraction.

Examples[edit | edit source]

Product Rule[edit | edit source]

Let where

Examples[edit | edit source]

Let Calculate

Differentiate both sides.

This shows that as above is valid when is a positive fraction.

Let . Calculate .

Differentiate both sides.

This shows that as above is valid for negative .

Let . Calculate .

Differentiate both sides.

This shows that as above is valid when is a negative fraction.

Quotient rule[edit | edit source]

Used where

Derivatives of trigonometric functions[edit | edit source]

sine(x)[edit | edit source]

The value :

>>> # python code
>>> [ (math.cos(dx)-1)/dx for dx in (.1,.01,.001,.0001,.0000_1,.0000_01,.0000_001,.0000_0001,.0000_0000_1) ]
[-0.049958347219742905, -0.004999958333473664, -0.0004999999583255033, 
-4.999999969612645e-05, -5.000000413701855e-06, -5.000444502911705e-07, 
-4.9960036108132044e-08, 0.0, 0.0]
>>>

by L'Hôpital's rule.

The value :

>>> # python code
>>> [ math.sin(dx)/dx for dx in (.1,.01,.001,.0001,.0000_1,.0000_01,.0000_001,.0000_0001,.0000_0000_1) ]
[0.9983341664682815, 0.9999833334166665, 0.9999998333333416, 0.9999999983333334, 
0.9999999999833332, 0.9999999999998334, 0.9999999999999983, 1.0, 1.0]
>>>

by L'Hôpital's rule.

Proof of 2 limits[edit | edit source]

Figure 1:

Area of sector area of triangle
Area of triangle area of sector


In the diagram

Let be the area of a sector of a circle. Then and

Area of sector

Area of triangle

Area of sector

Therefore

Therefore


cosine(x)[edit | edit source]

Differentiate both sides:

tan(x)[edit | edit source]

Differentiate both sides:

Derivatives of inverse trigonometric functions[edit | edit source]

arcsine(x)[edit | edit source]

Figure 2: Graph of and associated curves.


In the figure to the right you can see that the curves are the same curve. However curve is limited to

The derivative shows that the slope of is when and infinite when

arccosine(x)[edit | edit source]

Figure 3: Graph of and associated curves.

In the figure to the right you can see that the curves are the same curve. However curve is limited to

The derivative shows that the slope of is when and infinite when

arctan(x)[edit | edit source]

Figure 4: Graph of and associated curves.

In the figure to the right you can see that the curves are the same curve. However curve is limited to

The derivative shows that the slope of is when and when

Derivatives of logarithmic functions[edit | edit source]

[edit | edit source]

Consider the value specifically . L'Hôpital's rule cannot be used here because is what we are trying to find.

taken times.

We will look at the expression for different values of with (approx) in which case the expression becomes where is taken times.

>>> # python code
>>> N=Decimal(2)
>>> v2 = ( [ n for n in (N,) for p in range (0,70) for n in (n.sqrt(),) ][-1] -1 )*(2**70) ; v2
Decimal('0.69314718055994530941743560122437474084363865015406919942144')
>>> 
>>> N=Decimal(8)
>>> v8 = ( [ n for n in (N,) for p in range (0,70) for n in (n.sqrt(),) ][-1] -1 )*(2**70) ; v8
Decimal('2.07944154167983592825352768227031325913255072732801782513664')
>>> 
>>> N=Decimal(32)
>>> v32 = ( [ n for n in (N,) for p in range (0,70) for n in (n.sqrt(),) ][-1] -1 )*(2**70) ; v32
Decimal('3.46573590279972654709124760144583715956091114572543812435968')
>>> 
>>> N=Decimal(128)
>>> v128 = ( [ n for n in (N,) for p in range (0,70) for n in (n.sqrt(),) ][-1] -1 )*(2**70) ; v128
Decimal('4.85203026391961716593059535875094644210510807293198187102208')
>>>

Compare the values v8, v32, v128 with v2:

>>> v8/v2; v32/v2; v128/v2
Decimal('3.00000000000000000000176135549769209744528640235368520610520')
Decimal('5.00000000000000000000587118499230699148996545343403093128046')
Decimal('7.00000000000000000001232948848384468209997247971055570994695')
>>>

We know that . The values v2, v8, v32, v128 are behaving like logarithms. In fact is the natural logarithm of written as

Figure 5: Graphs of for a = .

Figure 5 contains graphs of for with graph of included for reference.

All values of are valid for all curves except where

The correct value of is:

>>> Decimal(2).ln()
Decimal('0.693147180559945309417232121458176568075500134360255254120680')
>>>

Our calculation produced:

Decimal('0.69314718055994530941743560122437474084363865015406919942144')

accurate to 21 places of decimals, not bad for one line of simple python code.

When

The base of natural logarithms is the value of that gives

This value of , usually called

>>> # python code
>>> N=e=Decimal(1).exp();N
Decimal('2.71828182845904523536028747135266249775724709369995957496697')
>>> ( [ n for n in (N,) for p in range (0,70) for n in (n.sqrt(),) ][-1] -1 )*(2**70)
Decimal('1.0000_0000_0000_0000_0000_0423_51647362715016770651530003719651328')
>>> # ln(e) = 1. Our calculation of ln(e) is accurate to 21 places of decimals.

When

ln(x)[edit | edit source]

Examples[edit | edit source]

Calculate

Careful manipulation of logarithms converts exponents into simple constants.

Chain rule[edit | edit source]

Used where

Examples[edit | edit source]

Let where .

Let where .

Applications of the Derivative[edit | edit source]

Shape of curves[edit | edit source]

The first derivative of or As shown above, at any point gives the slope of at point

is the slope of when

Figure 1: Diagram illustrating relationship between and
When both and slope of
When both and slope of
When both and slope of
Point is absolute minimum.

In the example to the right, and

Of special interest is the point at which or slope of

When and

The point is called a critical point or stationary point of Because has exactly one solution for has exactly one critical point.

The value of at point is less than at both Therefore the critical point is a minimum of

In this curve the point is both local minimum and absolute minimum.

Figure 1: Diagram illustrating relationship between and
When or both and slope of
When both and slope of
When both and slope of
When both and slope of
Point is local maximum.
Point is local minimum.

In the example to the right, and


Of special interest are the points at which or slope of

When and or

When and

The points are critical or stationary points of Because has exactly two real solutions for has exactly two critical points.


Slope of to the left of is positive and adjacent slope of to the right of is negative. Therefore point is local maximum. Point is not absolute maximum.


Adjacent slope of to the left of is negative and slope of to the right of is positive. Therefore point is local minimum. Point is not absolute minimum.

Maxima and Minima[edit | edit source]

Electric water heater[edit | edit source]

Figure 1(a): Graph of and for
with axis compressed.
For maximum and

A cylindrical water heater is standing on its base on a hard rubber pad that is a perfect thermal insulator. The vertical curved surface and the top are exposed to the free air. The design of the cylinder requires that the volume of the cylinder should be maximum for a given surface area exposed to the free air. What is the shape of the cylinder?

Let the height of the cylinder be and let where is the radius and is a constant.

Surface of cylinder

Volume of cylinder

For maximum volume,

Therefore

The height of the cylinder equals the radius.

Square and triangle[edit | edit source]

Figure 1(b): Graph of parabola with axis compressed.

A square of side has perimeter and area

An equilateral triangle of side has perimeter and area

Total area and must be minimum. What is the value of ?

For minimum

County Road[edit | edit source]

Figure 1(c): Plan of county road between Town A and Town B to be constructed so that cost is minimum.

Town B is 40 miles East and 50 miles North of Town A. The county is going to construct a road from Town A to Town B. Adjacent to Town A the cost to build a road is $500k/mile. Adjacent to Town B the cost to build a road is $200k/mile. The dividing line runs East-West 30 miles North of Town A. Calculate the position of point C so that the cost of the road from Town A to Town B is minimum.

Let point

Then distance from Town A to point

Distance from Town B to point

Figure 1(d): Graphs showing cost of county road and
Curve showing is actually .
Cost is minimum where

Cost in units of $100k.

For minimum cost

Cardboard box[edit | edit source]

Figure 1(e): Sheet of cardboard to be cut and folded to make box of maximum possible volume.
Cut on purple lines, fold on red lines.
Design of box includes top.

A piece of cardboard of length and width will be used to make a box with a top. Some waste will be cut out of the piece of cardboard and the remaining cardboard will be folded to make a box so that the volume of the box is maximum.

What is the height of the box?

Figure 1(f): Curves associated with design of cardboard box.
and is maximum when

For maximum volume

inches.

Rates of Change[edit | edit source]

The car jack[edit | edit source]

Figure 2: Photo of car jack illustrating horizontal and vertical rates of change.

In triangle to the right:

  • length inches,
  • length inches and is horizontal,
  • length inches and is vertical,

Point is moving towards point at the rate of inchesminute.

Vertical motion[edit | edit source]
Figure 3: Curves and values associated with car jack.
When
When
When

At what rate is point moving upwards:

(a) when ?

(b) when ?

(c) when ?

We have to calculate when is given.

(equation of circle)

For convenience we'll use the negative value of the square root and say that

Relative to line

When inchesminute.

When inchesminute.

When and inchesminute.

This example highlights the mechanical advantage of this simple but effective tool. When the top of the jack is low, it moves quickly. As the jack takes more and more weight, the top of the jack moves more slowly.

Change of area[edit | edit source]
Figure 4: Graph of and .

Area of is maximum when

At what rate is the area of changing when

(i) ?

(ii) ?

(iii) ?

where is area of and inches minute.

Calculate

When and area of is increasing at rate of square inches/minute.

When and area of is decreasing at rate of square inches/minute.

When is a maximum of square inches and

On the clock[edit | edit source]

Figure 5: Image of analog clock showing minute and hour hands at o'clock.

An old fashion analog clock with American style face (12 hours) keeps accurate time. The length of the minute hand is inches and the length of the hour hand is inches.

At what rate is the tip of the minute hand approaching the tip of the hour hand at 3 o'clock?

Let be distance between the two tips.

The task is to calculate

where is angle subtended by side at center of clock.

Calculating

Angular velocity of minute hand radians/hour.

Angular velocity of hour hand radians/hour.

angular velocity of minute hand relative to hour hand radians/hour.

Calculating

inches/hour.

Radar Speed Trap[edit | edit source]

Figure 6: Multi-lane highway oriented East-West.
feet.
feet.
.
mph.

A multi-lane highway is oriented East-West. A vehicle is moving in the inside lane from West to East. A law-enforcement officer with a radar gun is in position 50 feet South of the center of the inside lane. When the vehicle is 200 feet from the radar gun, it shows the vehicle's speed to be 60 mph. What is the actual speed of the vehicle?

Let length

where mph.


The derivative


mph.

On the Water[edit | edit source]

A ship is at sea nautical miles East and nautical miles South of a lighthouse, and the ship is steaming South-West at nautical miles/hour or knots.

At what rate is ship approaching lighthouse?

knots.

knots.

Figure 1: Diagram indicating position, course and speed of ship relative to lighthouse.

Calculating

The sign indicates that, when is increasing, is decreasing.

knots towards the lighthouse.

Reciprocating engine[edit | edit source]

Figure 1: Diagram illustrating mathematical relationship between piston, connecting rod and crankshaft in reciprocating engine.

The diagram illustrates the piston, connecting rod and crankshaft of an internal combustion reciprocating engine.

The connecting rod is connected to piston on axis, and to crankshaft at end of on axis.

This section analyzes the motion of the piston as the crankshaft rotates through angle and the piston moves up and down on the axis.

Position of piston[edit | edit source]
Figure 2: Diagram showing position of piston as a function of rotation of crankshaft in reciprocating engine.
Piston moves up and down between and inches.

Code supplied to grapher (without white space) is:

(5)(cos(x)) + ( ((25)((cos(x))^2) + 144 )^(0.5) )

When expressed in this way, it's easy to convert the code to python code:

(5)*(cos(x)) + ( ((25)*((cos(x))**2) + 144 )**(0.5) )

Positions of interest:

Velocity of piston[edit | edit source]
Figure 3: Diagram showing velocity of piston as a function of position of piston in reciprocating engine.
Strictly speaking, velocity
At constant RPM, is constant. Therefore is used to illustrate velocity.

Calculation of by implicit differentiation.

Acceleration of piston[edit | edit source]
Figure 4a: Diagram showing acceleration of piston in reciprocating engine.
Negative acceleration has a greater absolute value than the positive, but it does not last as long.

Acceleration introduces the second derivative. While velocity was the first derivative of position with respect to time, acceleration is the first derivative of velocity or the second derivative of position.

From velocity above

By implicit differentation:

Substitute for and as defined above, and you see the code input to grapher at top of diagram to right.

Figure 4b: Diagram showing "irregularities" in the curve of velocity while velocity is increasing.
axis compressed to illustrate shape of curves.

"Kinks" in the curve:

It is not obvious by looking at the curve of velocity that there are slight irregularities in the curve when velocity is increasing.

However, the irregularities are obvious in the curve of acceleration.

During one revolution of the crankshaft there is less time allocated for negative acceleration than for positive acceleration. Therefore, the maximum absolute value of negative acceleration is greater than the maximum value of positive acceleration.

Minimum and maximum velocity[edit | edit source]
Figure 5: Diagram showing positions of maximum velocity of piston in reciprocating engine.
In the first quadrant, from to the piston moves through inches.
In the second quadrant, from to the piston moves through inches.
Therefore, in the first quadrant, acceleration must be greater than in the second quadrant.

Velocity is rate of change of position. See also Figure 3 above.


Minimum velocity:

Velocity is zero when slope of curve of position is zero. This occurs at top dead center and at bottom dead center, ie, when and


Maximum velocity:

Intuition suggests that the position of maximum velocity might be the point at which the connecting rod is tangent to the circle of the crankshaft. In other words:

or, that the position of maximum velocity might be the point at which the piston is half-way between top dead center and bottom dead center. In other words:


However, velocity is maximum when acceleration is which occurs when

Suppose that the engine is rotating at radians/second or approx. RPM.

abs inches/second or approx. mph.

Minimum and maximum acceleration[edit | edit source]
Figure 6: Diagram showing positions of minimum and maximum acceleration of piston in reciprocating engine.

Acceleration is rate of change of velocity. See also Figures 4a and 4b above.


Minimum acceleration:

Acceleration is zero when slope of curve of velocity is zero. This occurs at maximum velocity or when is approx.


Maximum acceleration:

Acceleration is maximum when slope of curve of velocity is maximum.

Maximum negative acceleration occurs when slope of curve of velocity is maximum negative. This happens at top dead center when

Maximum positive acceleration occurs when slope of curve of velocity is maximum positive. This happens before and after bottom dead center when is approx.


Let the engine continue to rotate at radians/second.

abs inches/second or approx. times the acceleration due to terrestrial gravity.

This maximum value of acceleration is maximum negative when


According to Newtonian physics , force = mass*acceleration, and , work = force*distance. In this engine energy expended in just accelerating piston to maximum velocity is proportional to rpm.

Perhaps this helps to explain why a big marine diesel engine rotating at low RPM can achieve efficiency of

Simple laws of motion[edit | edit source]

Let a body move in accordance with the following function of where means time:

where is position at time

has the dimension of length. Therefore, each component of must have the dimension of length.

For to have the dimension of length, must have the dimensions of or velocity.

For to have the dimension of length, must have the dimensions of or acceleration.

If and

The derivatives enable us to assign specific values to

where is velocity at time

If and

a constant equal to the acceleration to which the body is subjected.

For convenience let us say that where is the (constant) acceleration to which the body is subjected.

Then and


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