firmfunda
  maths > differential-calculus

Differentiation: First Principles


    what you'll learn...

Overview

Differentiation : First Principles

 »  cause-effect relation in two quantities

    →  eg: speed-displacement

 »  The effect is calculated as a function of an algebraic expression in a variable.

    →  eg: displacement s=3t2+2

 »  The cause is derived to be "rate of change of effect with respect to the variable".

    →  eg : Speed = rate of change of displacement

 »  In such a case, the cause is another algebraic expression in the variable.

    →  speed is a function of t

 »  The cause is computed as rate of change : the change in effect for a small change δ0 in the variable.

    →  speed =limδ0s(t+δ)-s(t)δ

 »  Derivative or Differentiation of a function

    For a small change in variable x, the rate of change in the function f(x) is the derivative of the function.

ddxf(x)

=f(x)

=limδ0f(x+δ)-f(x)δ

rate of change

A car travels a distance 20m in 2 seconds. The speed of the car is "20m/2sec=10 m/sec"

Speed is computed as distance traveled (change) per unit time (rate).

Speed is rate of change of distance.

quantity as an expression

A car is moving at speed 20m/sec. What is the distance covered in t sec ?

without a numerical value for t the distance traveled can be given as an expression.

For any value of t, the distance traveled is s=20t meter

Measurement can be expressed as a function of a variable.

rate of change of expression

A car is moving with displacement given as a function of time s=3t2+2. The distance at t=2sec is computed by substituting t=2 in s=3t2+2 =3×22+2=14m

A car is moving with displacement given as a function of time s=3t2+2. Couple of students want to calculate the speed.
We know that speed=distancetime

 •  Person A does the following: At t=2s, the distance traveled is 14m. So the speed is 142=7m/sec.

 •  Person B does the following: At t=1s, the distance traveled is 5m. So the speed is 51=5 m/sec.

Which one is correct? "Neither of them". The answer is explained in the following.

Person A calculated average speed till 2sec

Person B calculated average speed till 1sec"

The results from these two are different and we understand that the average speed changes with time and we expect that "speed is a function of time".

Speed is rate of change of displacement. So, one has to figure out how to find the rate of change of an algebraic expression.

instantaneous

Consider the problem, a car is moving with displacement given as a function of time s=3t2+2. Speed is the rate of change. Few students are set to find rate of change


 •  Person A found rate of change for 1 second interval. 3(t+1)2+2-3t2-2 =6t+3

 •  Person B found rate of change for 2 second interval. 3(t+2)2+2-3t2-22 =6t+6

 •  Person C found rate of change for 0.5 second interval 3(t+.5)2-3t2.5 =6t+1.5

The above actually provides "Average of two values representing speed at t and t+δ". That is, for Person A, the δ=1, for person B, the δ=2, and for person C, the δ=0.5.

The calculation does not provide instantaneous speed. What is "instantaneous".


A speedometer is attached to a wheel of the car. The speedometer measures the speed at which the wheel rotates and proportionally provides the speed of the car. The speed shown in the speedometer is the "instantaneous speed for any given time".

when interval is 0

The average of speed between t and t+δ is s(t+δ)-s(t)δ =6t+3δ. If the value of δ is small, then the accuracy of the instantaneous speed is better.

Instantaneous speed is when the time interval δ is zero. This is a big step in understanding. The displacement is continuously changing with time and at an instance the rate of change is calculated.

substitute 0

The instantaneous speed is s(t+δ)-s(t)δδ=0. The value of this calculation when δ=0 is substituted is "indeterminate value 0/0".

To solve a function evaluating to indeterminate value "Use Limit of the function as δ approaching 0."

limit

A car is moving with displacement given as a function of time s(t)=3t2+2. The instantaneous speed is
s(t+δ)-s(t)δδ=0
=s(t)-s(t)δ
=00

Since the speed evaluates to indeterminate value, the limit is used:

limδ0s(t+δ)-s(t)δ

    =limδ03(t+δ)2+2-3t2-2δ

canceling 2 and -2 and expanding the square
    =limδ03t2+6tδ+3δ2-3t2δ

canceling 3t2 and -3t2
    =limδ06tδ+3δ2δ

canceling δ from numerator and denominator
    =limδ06t+3δ

applying limit.
    =6t


The instantaneous speed is computed as an algebraic expression.

summary

Summarizing the learning so far:

 •  Two quantities are in a cause-effect relation.

 •  The effect is calculated as a function of an algebraic expression in a variable.

 •  The cause is derived to be "rate of change of effect with respect to the variable".
(note: there are other forms of relation between cause-effect, such as multiple, addition, exponent. In this topic, we are concerned with only the rate of change relation.)

 •  In such a case, the cause is another algebraic expression in the variable.

 •  The cause is computed as rate of change : the change in effect for a small change δ0 in the variable.

This calculation is named as differentiation or derivative of the function.

when to?

Differentiation in the context of cause-effect relation: If effect is given by f(x) and the cause is the rate-of-change, then the cause is computed as differentiation or derivative of f(x) denoted as ddxf(x) or f(x).

ddxf(x)=limδ0f(x+δ)-f(x)δ. It is duly noted that the cause-effect can be of various relationships, like sum, difference, product, etc. Only when the cause-effect is established to be rate-of-change, the differentiation is used to get instantaneous-rate-of-change.

Cause-effect is explained to understand the physical significance. Abstracting this and understanding the quantities involved in differentiation: A quantity u=f(x) is related to another quantity v such that v is the instantaneous rate of change of u with respect to x, then
v=dudx=limδ0f(x+δ)-f(x)δ

Note that dudx is another quantity related to the given quantity u.

various forms

The derivative of the function y=f(x) can be given in different forms:

For a small change in variable x, the instantaneous rate of change of the function y=f(x) is the derivative of the function.

dydx

=y

=limΔx0ΔyΔx

=ddxf(x)

=f(x)

=limδ0f(x+δ)-f(x)δ

=limΔx0f(x+Δx)-f(x)Δx

The instantaneous rate of change of a function is called "differentiation or derivative of the function".


Students can connect the notation ddxf(x) as

 •  the small difference in x is given by denominator dx

 •  the effective difference (because of small difference in x) in the function is given by numerator d

 •  ddx denotes the difference in function with respect to a small difference in x.


The instantaneous rate of change of a function with respect to the variable is the derivative of the function. The word "instantaneous" may be dropped and it is often referred to as "rate of change".

examples

Finding the derivative of y=x2+x in first principles:

dydx

=limδ0[f(x+δ)-f(x)]/δ

=limδ0[(x+δ)2+x+δ-(x2+x)]/δ

=limδ0[x2+2δx+δ2+x+δ-x2-x]/δ

=limδ0[2δx+δ2+δ]/δ

=limδ02x+δ+1

Substituting δ=0,
ddx(x2+x)=2x+1


Finding the derivative of y=sinx in first principles:

dydx

=limδ0[f(x+δ)-f(x)]/δ

=limδ0[sin(x+δ)-sinx]/δ

=limδ0[sinxcosδ+cosxsinδ-sinx]/δ

=limδ0cosxsinδδ-limδ0sinx-sinxcosδδ

=cosxlimδ0sinδδ-sinxlimδ01-cosδδ

applying the standard limits
=cosx×1-sinx×0

That is
ddxsinx=cosx"

summary

Derivative or Differentiation of a function : For a small change in variable x, the instantaneous rate of change in the function f(x) is the derivative of the function.

ddxf(x)

=f(x)

=limδ0f(x+δ)-f(x)δ

Outline

The outline of material to learn "Differential Calculus" is as follows.

•   Detailed outline of Differential Calculus

    →   Application Scenario

    →   Differentiation in First Principles

    →   Graphical Meaning of Differentiation

    →   Differntiability

    →   Algebra of Derivatives

    →   Standard Results