On the solution of left derivative and right derivative Let f (x) = x / (1 + e ^ (1 / x)) x not equal to 0, f (x) = 0, x = 0, then what is the left derivative and the right derivative equal to

On the solution of left derivative and right derivative Let f (x) = x / (1 + e ^ (1 / x)) x not equal to 0, f (x) = 0, x = 0, then what is the left derivative and the right derivative equal to

F (x) '= limx tends to 0 [x / 1 + e ^ 1 / x-f (0)] / (x-0) = Lim1 / (1 + e ^ 1 / x), right derivative, X tends to 0 +, denominator tends to infinity, the whole tends to 0; left derivative, x tends to 0 -, denominator tends to 1, the whole tends to 1

On the solution of derivative
How can f (x) g (x) be derived from x
For example, xexp (2x) is derived from X

(f(x)g(x) )'=f'(x)g(x)+f(x)g'(x)
You can take this example
(xexp(2x))'=x'(exp(2x))+x(exp(2x))'=exp(2x)+2x*exp(2x)=(2x+1)exp(2x)

1 / 2,
1 / 2 of 2,
One third of 2,
……
One nth power of 2

Sn=1/2^1+1/2^2+1/2^3+…… 1/2^n
(1/2)Sn= 1/2^2+1/2^3+…… 1/2^n+1/2^(n+1)
So Sn - (1 / 2) Sn = 1 / 2 ^ 1-1 / 2 ^ (n + 1)
So Sn = 1-1 / 2 ^ n