It is often the case that we need to differentiate a simple function of a simple function, such a exp(sqrt(x)).
We may know how to differentiate sqrt(x), and how to differentiate exp(x), but differentiating the combination is not in our common knowledge nor is it readily available in common math tables. ... What do we do?
The CHAIN RULE comes to our rescue. It provides us with a way to differentiate a complicated function using our knowledge of how to differentiate the simpler component subfunctions from which it is constructed.
chain rule states that the derivative of a function of g, with respect to x is equal to the derivative of our function of g, with respect to g, times the derivative of g with respect to x.
An application of the chain rule which is particularly important in Quantum Physics is the the case where we need to find the derivative of an exponential function of a function of x. Here the convenient solution is that the derivative is the original function times the derivative of the function of x which appears as the exponent of e.