A question on differentiation of exponential function Recently,one of my friend asked me a question but I'm not able to answer this question .His question about differentiation of exponential function .Here is his question:  

 A: Let $f(x) = e^x$. This might be for example the size of a population of bacteria multiplying by a factor $e$ every hour. It stands to reason that the rate at which the bacteria are increasing is proportional to the amount present. (Each unit mass of bacteria is responsible for the same amount of increase.)
That explains why $y'$ is proportional to $y$, but not why it's equal. If $f'(0) = 1$, then the constant of proportionality will be $1$ and we will have equality. This is something specific to the number $e$, and in fact the number $e$ was chosen precisely so that this would happen. 
How can $e$ be chosen for this to happen? If you have an exponential function, say $g(x) = 2^x$, compress or expand the graph horizontally so that the slope at $(0,1)$ becomes $1$. Then we have the graph of a function $f(x) = g(cx)$ for some constant $c$. But $f(x) = 2^{cx} = (2^c)^x$. So it's enough to define $e = 2^c$.
A: The value of $e^x$ at a point, $x_0$ is also the same as the slope at that point. So $\frac{d}{dx}e^{x_0}=e^{x_0}$
