If f is differentiable everywhere, then, in terms of f and a, what is the value of: lim x-->a (f(f(x))-f(f(a)))/(x-a)?
That equation simply finds the slope of a secant line between the points (x,f(x)) to (a,f(a)) as x approaches a. When x approaches a, the slope of the secant line begins to approach the slop of f(x) at a. In other words, it finds the slope at a or the derivative at a. Therefore, it equals f'(a).
I'm afraid not. Unless the f(f(x)) and f(f(a)) are typos, the answer is the derivative of f(f(x)) at x = a, NOT the derivative of f(x) at x = a ......
Using the chain rule:
. At x = a, this will equal . Making it a little neater, if you knew that f(a) = b, then it will equal ......