More on composite functions

We have seen that there are various uncertainties in our derivation of the formula for the derivative of a composite function, and that there are two main issues. One is that it we have no way to know if we are dividing by at one stage in our computation, and another is that we have a limit which appears to be a derivative, but does not actually appear in the form of our definition of derivative. We now will proceed to evade these difficulties.