How do we use the Chain-rule when differentiating?

The Chain-rule is used to differentiate a function of a function. Running through an example:Let's say want to differentiate the function y = (2x+1)^3. We can substitute a variable, w, in order to differentiate using the chain rule.Let (2x+1) = w . We now have y = (w)^3. If we differentiate this we get the function differentiated with respect to w.Hence, (dy/dw) = 3(w)^2. However we want to find (dy/dx). If we look carefully we can see that (dy/dx) = (dy/dw) * (dw/dx). This means that if we can find (dw/dx), we can multiply it by 3(w)^2 to find (dy/dx). We know that w = 2x+1. Differentiating this with respect to x we get (dw/dx) = 2. Now we have what we need to find (dy/dx).(dy/dx) = (dy/dw) * (dw/dx)(dy/dx) = [3(w)^2] * [2] (dy/dx) = 6(w)^2.Now all we need is to put it in terms of x. If we substitute w = 2x+1 into (dy/dx) we get:(dy/dx) = 6(2x+1)^2.This is our final answer. In general, we can find (dy/dx) using the chain rule by applying the fact that (dy/dx) = (dy/dw)(dw/dx).For functions of functions of functions we can use (dy/dx) = (dy/dw)(dw/du)*(du/dx) and so on for longer functions.