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JEE Main: Differentiation and How To Calculate ‘e’
Edited By Ramraj Saini | Updated on Apr 19, 2024 07:49 PM IST | #JEE Main
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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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To pursue Bachelor of Technology (B.Tech) after 12th, you need to appear in JEE MAINS, a National level entrance test conducted by NTA twice a year to offer candidates in B.Tech course.
If you are not prepared for JEE MAINS, you can appear in the AP EAMCET which is state level entrance test conducted in Andhra Pradesh to offer admission to B.Tech course in the Top engineering colleges in Andhra pradesh through AP EAMCET.
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Devraj 12 January,2025
Hello aspirant,
No, you cannot change the language of the JEE Main exam during the examination. Here are the key points:
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Hope it helps !
Prachi Kumari 12 January,2025
JEE exam is conducted by NTA for admission to top IITs and NITs for different courses. Based on the previous year trends, for 2025,jee exam, students can expect the cut off between 90-95 percentile for general category students and 80-85 for OBC category and 55-60 for reserved categories. To know more,visit JEE cut off 2025
Annu Kumari 11 January,2025
Ofcourse you can give both the exam JEE and GUJCET as JEE exam give admission to IIT, NIT, and GFTIs for B. Tech courses and GUJCET offers seats in the universities situated in Gujarat.by giving two exams you have more options of colleges to choose from for your b.tech course.
If you see Engineering Admission Process are like this
Josaa: This is JEE counselling process. This would Only admitted students on the basis of JEE Mains marks and would be considered for your rank.
ACPC: This is your state admission process which GUJCET marks along with PCM boards theory marks would be considered. 50% of college seats are given to ACPC.
Since you are interested in both JEE as well as GUJCET, I'd advice you to enroll through both the processes for good colleges and good branches.
Best of luck!!!
sadafreen356 10 January,2025
Hi,
To be eligible for the admission in Thapar University B. Tech programme, candidates must qualify the JEE Main exam, which will be followed by counselling and seat allocation.
The admission in Thapar University is done on the basis of the JEE Main rankings and category-wise cutoffs will be released separately for all the B. Tech courses.
To be eligible for JEE MAINS, candidate must passed 10+2 with minimum of 60% marks from a recognized board. Appearing students are also eligible for JEE MAINS.
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