Differentiation - Demystified

All of us might have studied differentiation. But how many of us like this?
The moment we hear differentiation most of us will remember about d/dx. In school days we used to byheart the differential of many terms and use it for calculations.

What is this differentiation all about, why it is needed and what is it practical application?

Let me try to demystify DIFFERENTIATION through the following questions and answers. Do read it and let me know whether this helped to make your understanding of differentiation more clear.

Is there any relation between differentiation and difference?
Indeed yes, differentiation is infact derived from difference. The relationship between these two will be evident from some of the other questions here.

Why is differentiation needed in real world?
Assume that a huge block of ice is melting. Depending on the atmospheric temperature the rate at which ice melts will vary. Can we compute the time required for the melting of ice without actually waiting for the whole ice block to melt.

Suppose we are heating one end of a long iron rod. Can we compute the heat the other end of the iron rod without actually measuring the temperature at the other end, or can we measure the temperature at any given point of the rod?

A rocket needs to be launched. Can we compute where it will be, once the accelaration of the rocket is known.
All the above computations are possible if we know differentiation.

What is differentiation?

Differentiation represents rate of change of any quantity with respect to another quantity.

What does the above sentence means?

Consider the following table, quantity1 and quantity 2 is varying as shown below

quantity1	quantity2
1	2
2	4
3	6
4	8
5	10

Quantity1 is changing by a unit of 1. (How did I get this - -> take the difference of adjacent values)
Quantity2 is changing by a unit of 2. (How did I get this --. take the difference between adjacent values)

What is the rate of change of quantity2 with respect to quantity1 or

if the question is phrased in another manner
How much will quantity2 change when quantity1's value is changed by a unit value?
To find this take any two adjacent values of quantity1 and its corresponding values in quantity2.
Let us take the first two values -> quanity1 - 1 & 2, quantity2 -> 2 & 4
Rate of change of quantity2 with respect to quanity1 = (4 - 2)/(2 - 1) = 2;
take the next set of values, computing similarly we will get = ( 6 - 4 ) / ( 3 - 2 ) = 2

If you take any set of value, you can see the rate of change is 2.
Even though the values are changing, the rate of change is a constant here.

So for the above set of quantities we found out the rate of change.

This rate of change is what is called differential.

Here we found out the differential by taking the ratio of difference of two sets of values.

In mathematics, quantity2 can be represented as a function of quantity1.
Watch the set of numbers of quanity2 and there corresponding numbers in quantity1.
it can easily be found out that all the values satisfied the following relation:

quantity2 = 2 x quantity1, if quantity2 is denoted by 'y' and quantity1 by 'x', then we can say that

y = 2*x

We had found out differential of quantity 2 with respect to quantity1,

ie differential of y with respect to x.

This is denoted by dy/dx.

This means rate of change of y with respect to x.

Here it was shown that when y = 2x, the differential - dy/dx = 2.

Another example of differentiation

consider the function y = 2,

this means that the value of y will be 2, whatever be the value of x,

so our earlier table will look like this

x	y
1	2
2	2
3	2
4	2
5	2

dy/dx = (2-2)/(2 - 1) = 0

Take any pair of values, the differential of y with respect to x will be zero.

This means that when y = 2, dy/dx = 0.

The examples given above shows that, whenever one quantity varies its value with respect to another the rate of change of quantity with respect to another quantity can also be represented as a function.

for the function y = 2x

differential is 2,

for the function y = 2

differential is 0

The underlying principle of differentiation shows that the difference in values of one quantity with respect to another is called differentiation.

Purpose of taking differentiation can be summarized as follows

Most of the real world problems can be expressed as functions.

To examine the rate of change of a function, we have to take the differential of the function.

The resultant equation after differentiation gives an expression which corresponds to rate of change of the function.

There may be cases where we take the differential of the result of differential of a function.

This is needed to find out the rate of change of "rate of change" of a function.

y = f(x) -> y is a function of x, ie y varies when x varies according to the relation.

dy/dx = d f(x)/dy - differential of y with respect to x, gives the rate of change of y with respect to x.

Once this is computed the resultant differential can be used to find out the rate of change of y for any given value of x, easily.

How will differentiation help in finding out the heat at a particular distance of iron rod whose one end is heated?

Let us assume that we know the equation for heat transfer as a function of distance from the heat source.

Differentiate this equation with respect to distance.

The result of this differentiation will the equation corresponding to the rate of change of heat at any given distance from the source.

Using this equation we can easily compute the heat at at particular point away from the heat source.

Hence differentiation of a function helps us to realise equations corresponding to the rate of change of the function. The resultant equations can be used to solve pratical problems concerned with rate changes.