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JEE Main: Differentiation and How To Calculate ‘e’
Updated on Apr 19, 2024 07:49 PM IST | #JEE Main
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) = ex, 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) = 2x.
Using First Principle of Differentiation, we know that the differentiation of a function f(x) is
So, differentiation of f(x) = 2x will be
Now, when the value of the limit h→0 [(2h-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 [(2h-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, [(2h-1)/h] = 0.6933…
When h = 0.0001, [(2h-1)/h] = 0.6931…
When h = 0.00000001, [(2h-1)/h] = 0.6931…
We can see that [(2h-1)/h] value approaches 0.6931…and hence
limit h→0 [(2h-1)/h]= 0.6931…
From equation (i):
f'(x) = 2x.(0.6931…)
So, differentiation of f(x) = 2x is of the form
f'(x) = Some constant. f(x)
Now if we do the same procedure with f(x) = 4x
When h = 0.001, [(4h-1)/h] = 1.3872…..
When h = 0.0001, [(4h-1)/h] = 1.38631…
When h = 0.00000001, [(4h-1)/h] = 1.38629…
So, limit h→0 [(4h-1)/h] = 1.38629…
And from (ii),
f'(x) = 4x . (1.38629…)
So, differentiation of f(x) = 4x 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) = ax equals some constant times ax
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 ax is 1. ax, meaning that the differentiation of the function ax is this function itself ( = ax)
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(ex)= ex
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 eiπ = - 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…).
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You can participate in CSAB Special Rounds after JoSAA if any lower NIT/IIIT or GFTI has CSE or related branches open for OBC-NCL + Female category.
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Chances are very slim at your rank even in CSAB.
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Try CUET UG 2025 for reputed Central Universities CSE programs.
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Rohan 14 May,2025
It is improbable that you would get admitted to SIT Lonavala's Computer Science and Engineering program with a JEE Main score of 62 percentile. CSE cutoffs have been far higher in other years, with closing ranks equal to percentiles exceeding 85–89. Other fields or universities with lower cutoffs could be worth looking into.
Swayam 13 May,2025
Sure, a jee main marks of 62.6% with 12 pass, you will admission in Sinhgad College of Engineering, Lonavala.
With you jee main marks you will be considered in All india quota or management quota. Also for some top branches you cannot choose with this mark but you can go with other branches like CS , civil etc..
All the best!!!
Admission with first class in JEE
Rubina Naseer 13 May,2025
With a 72 percentile in JEE Main 2025 and an expected 80%+ in your 12th board exams , securing admission to B.Tech in Computer Science & Engineering (CSE) at Guru Nanak Dev University (GNDU), Amritsar may be challenging.
A 72 percentile typically translates to an AIR between 250,000–300,000 , which is significantly higher than the previous year's closing ranks for CSE and related branches at GNDU.
- Consider applying for branches with higher closing ranks, such as Computer Engineering or Cyber Security , though admission may still be competitive.
- Look into other universities or colleges
- For more detailed information, you can visit GNDU's official admission page: GNDU Admissions (https://www.gnduadmissions.org/)
Rohan 13 May,2025
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