if first derivative is the slop of the given functions, then what is the physical meaning of exponential function remaining the same function after differentiation?? does it mean its vertical tangency make it indifferentiable? plz clarify me the concept... regards
It means that the curve of the exponential function has the same instantaneous rate of change at a given point as its value.
I'm afraid you will have to tell us what you mean by the "physical meaning" of a mathematics statement.
It means that the rate of change at a given point is the same as the value of the function at that point. So, what does this mean physically? Suppose I have a population of things that reproduce -- people on earth, bacteria in a dish, whatever. Since they're reproducing like crazy, the number of new individuals in any given interval of time is proportional to how many individuals there already are. If there are lots of individuals, then there will be lots of new individuals made. That's why the exponential function is intimately involved in the growth of populations. Same thing with compound interest. The amount of interest you get is proportional to how much money you already have. And the formula for compound interest does in fact turn out to be an exponential function. That's the physical meaning. The amount of growth (the derivative) is proportional to the amount of stuff that's already there.