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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
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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Have a question related to JEE Main ?
Hello Aastha,
Yes, it's okay to use your mother's name as per the Aadhaar card, "Mamta Walia," while filling out the JEE application form. However, during document verification or counseling, there could be a discrepancy between the 10th marksheet and Aadhaar card. To avoid future issues, keep an affidavit or a notarized document ready explaining the name difference, or get it corrected in either document if possible.
For more details, you can check out an article by Careers360 whose link is given below:
Link: https://engineering.careers360.com/articles/how-to-fill-jee-main-application-form
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
kumardurgesh1802 16 November,2024
Yes, correction window allow the changes photo uploaded along with educational or personal details etc. The scanned image should be of size 4 kb to 30 kb. The changes that are not allowed in correction window is change of exam centre.
Correction window open dates will be announced by NTA on their official website.
For more details about jee mains correction, you can visit career 360 website
Annu Kumari 16 November,2024
Hello,
Honestly, no, it's not possible to appear for the January JEE session as a General candidate and the April JEE session as a PwD candidate.
As per the NTA guidelines, candidates can appear for JEE Main only once in a year with the same category and eligibility criteria. If you've already appeared for the January session as a General candidate, you cannot change your category to PwD for the April session.
Moreover, PwD candidates need to provide valid disability certificates and other supporting documents during the application process, which cannot be changed later.
So, it's essential to choose your category and eligibility criteria carefully before applying for JEE Main.
https://engineering.careers360.com/exams/jee-main
Hope this helps,
Thank you
Tanya Gupta 15 November,2024
Hello aspirant,
The JEE Mains 2025 test will be administered by the National Testing Agency in two sessions in January and April of 2025. Students need to devote more time to their JEE Main preparation as the JEE Main 2025 session 1 exam is just a few months away. It is wise to work efficiently and prepare in an effective manner. NTA JEE is unquestionably one of the hardest tests to pass, however students can achieve exceptionally high scores if they prepare properly.
To know the complete preparation strategy, you can visit our site through following link:
https://engineering.careers360.com/articles/jee-main-preparation-tips
Thank you
Sajal Trivedi 15 November,2024
Students who completed 10+3 can appear for JEE Main and can also be eligible for Delhi University, but their rank and score are not applicable for admission to NITs and CFTIs.
Moreover,Some B.Tech programs at DU do not require a JEE Mains scorecard. For example, you can secure admission to the B. Tech in Information Technology and Mathematical Innovation program by taking the Common University Entrance Test (CUET UG) exam.
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