"Inside function" and "outside function : Generally we don't actually do all the composition stuff in using the Chain Rule. That can get little complexes and actually obscures the fact that there is a quick & easy way of remembering the chain rule which doesn't need us to think in terms of function composition.

Let's take the following function

This function contain an "inside function" & an "outside function". The outside function is square root/ the exponent of  ½ based on how you desire to think of it and the inside function is the stuff that we're taking the square root of or raising to the  1 , again based o how you desire to look at it.

Then the derivative is,

Generally it is how we think of the chain rule.  We recognize the "inside function" & the "outside function". Then we differentiate the outside function leaving the inside function alone & multiply all of this by the derivative of the inside function.  General form of this is following,

We can always identify the "outside function" in the examples below by asking ourselves how we would evaluate the function.  In the R(z) case if we were to ask ourselves what R(2)

is we would primary evaluate the stuff under the radical and then finally take the square root of thisresult. The square root is the last operation that we perform in the evaluation and this is also the outside function. The outside function will for all time be the last operation you would perform if you were going to evaluate the function.