JEE Main: Differentiation and How To Calculate ‘e’

We all have come across the number ‘e’ while studying logarithm and calculus. This number is an irrational number like π, and is called Euler’s Number. Its value is 2.71828… and this sequence of digits never ends. So, how can we find this number and what is so special about this number?

A property that you must have studied in differentiation that stands out among all differentiation formulae is

So, the differentiation of the function f(x) = e^x, is this function itself. This makes this function unique. We will be using this property to calculate the number ‘e’. So, in all the calculations below, we will not use ‘e’ directly.

Let us start with the differentiation of a general exponential function, f(x) = 2^x.

Using First Principle of Differentiation, we know that the differentiation of a function f(x) is

So, differentiation of f(x) = 2^x will be

Now, when the value of the limit _h→0 [(2^h-1)/h] is calculated by putting different values of h that are very close to 0, we can see that the value of this expression approaches 0.693147…(Note that we are not directly using the value of this limit as ln(2), as the number ‘e’ and thus ln(x), which is log with the base ‘e’, is not yet known).

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Let us see what values of [(2^h-1)/h] we get by putting different values of h that are close to 0. We can use a calculator to find these values

When h = 0.001, [(2^h-1)/h] = 0.6933…

When h = 0.0001, [(2^h-1)/h] = 0.6931…

When h = 0.00000001, [(2^h-1)/h] = 0.6931…

We can see that [(2^h-1)/h] value approaches 0.6931…and hence

limit _h→0 [(2^h-1)/h]= 0.6931…

From equation (i):

f'(x) = 2^x.(0.6931…)

So, differentiation of f(x) = 2^x is of the form

f'(x) =  Some constant. f(x)

Now if we do the same procedure with f(x) = 4^x

When h = 0.001, [(4^h-1)/h] = 1.3872…..

When h = 0.0001, [(4^h-1)/h] = 1.38631…

When h = 0.00000001, [(4^h-1)/h] = 1.38629…

So, limit _h→0 [(4^h-1)/h] = 1.38629…

And from (ii),

f'(x) =  4^x . (1.38629…)

So, differentiation of f(x) = 4^x is again of the form

f'(x) =  Some constant. f(x)

In fact, we can do the same exercise for any positive real number a, and we will find that the differentiation of f(x) = a^x equals some constant times a^x

It can also be seen that the value of this constant keeps on increasing as the value of ‘a’ increases. For example

For a = 2, the constant we calculated was 0.6931…

For a = 4, the constant we calculated was 1.38629…

Similarly, for a = 5, the constant can be calculated to be 1.6094…

For a = 6, the constant is 1.7917…

So naturally we can ask ourselves the question that can we find a number ‘a’ for which this constant value equals 1, and thus differentiation of a^x is 1. a^x, meaning that the differentiation of the function a^x is this function itself ( = a^x)

After doing multiple hits and trials, this number can be found to be 2.71828. That is why we have the unique property,

d/dx(e^x)= e^x

Euler’s Number also finds applications in fields of mathematics other than calculus. One of the most important applications is in Complex Numbers. You must have come across the relation e^iπ = - 1. Imaginary powers of e help us get the values of many trigonometric series which would otherwise be very difficult to prove using only the trigonometric relations. The number is also used in Finance (to calculate compound interest), to explain population growth of humans or microbes, to explain radioactive decay (which in turn is used to tell the age of ancient objects), etc.

Due to numerous applications, ‘e’ is the second most famous mathematical constant after π. We also celebrate ‘e-day’ on 7 February. This date is chosen as it is written as 2/7 in month/date format and the digits 2,7 represent the first two digits used in the value of ‘e’ (2.71…).