Base of differential fixed points

Consider the definition of the difference quotient in terms of the quotient produced by an interval of length $h$. $$\frac{f(x+h)-f(x)}{h}$$ We want to study the fixed points of this difference quotient, supposing that $h$ is fixed. In order to do so we can equate $h$ with $f(x)$. To simplify this equation we can multiply both sides by $h$ and then add $f(x)$ to both sides. $$\frac{f(x+h)-f(x)}{h}=f(x)$$ $$f(x+h)-f(x)=hf(x)$$ $$f(x+h)=(h+1)f(x)$$ At this point we can raise $h+1$ to the power of $1/h$ to get the base of the fixed point of difference quotient with fixed length $h$. $$b = (h+1)^{\frac{1}{h}}$$ This formula works in general to compute the base of the fixed point of a difference quotient of any given length. We tend to want to reduce our consideration to the case of unit fractions $1/n$ in which case we get the following familiar formula for the base of the differential of the fixed point of a given unit fraction length: $${(\frac{n+1}{n})}^n$$ Actual values of this sequence of rational numbers are provided below. The first case $2$ is the fixed point of the difference quotients of length $2$ or the difference quotients of length $1$, the next number $9/4$ is the base of the difference quotients of length $1/2$, and the next is the base of the difference quotients of length $1/3$, and so on. $$2, \frac{9}{4}, \frac{64}{27}, \frac{625}{256}, \frac{7776}{3125} ...$$ As a result, in the case of finite differences it is possible to use $2^x$ as a fixed point. The consecutive differences of the $2^x$ sequence are the values of $2^x$ itself. But this $2^x$ value does not suffice when taking difference quotients of smaller lengths. $$1,2,4,8,16,32,64,128,256,...$$ In order to get the fixed point of the difference quotients of smaller lengths it is necessary to use the formula ${(\frac{n+1}{n})}^n$ we derived earlier to get a different base. If we then continue this process to infinity we get the base of the fixed point of the derivative which is $e$ which leads to the function $e^x$ which is equal to itself under differentiation. This demonstrates that the limit ${(\frac{n+1}{n})}^n$ is not simply a random means of generating $e$ it is actually a formula for the different fixed points of the difference quotient. So actually the identity of $e$ is as the fixed point of the derivative and the limit definition is just computing different fixed points of the difference quotient.