Mr Tambuwal, thank you for proving the derivative of e^x by first principles without using circular reasoning like many videos do. Most other videos prove this using L’Hospitals’ rule or Taylor expansion which doesn’t make sense from a logic point as they are assuming the thing they are trying to prove. Please continue your amazing work :)
This can be shown easily. Lim (h->0) (e^h - 1)/h is 1 can be showed by expanding e^h = 1 + h + h^2/2! + ... so that e^h - 1 = h + h^2/2! + h^3/3! + .... and hence (e^h - 1)/h = 1 + h/2! + h^2/3! + .... and consequently the limiting value of this expression when h -> 0 is 1 as the rest of the terms go to 0.
You cannot use this method. Because the Taylor expansion involves knowledge on the derivative of e^x in the first place. So it is circular reasoning and is incorrect.
