differentiate y = (4-x)^2

This is a basic example of a very important result: the chain rule. The difficulty of this sort of example is that we have a "function of a function". That is, we have the function '4-x' and then we square this. 

The general approach is as follows. First we will let 'u' be a new function: u=4-x. It is evident that now we have y=u^2 which looks like it might be easier to work with. The chain rule says the following:

dy/dx = (dy/du)*(du/dx)

In this case y=u^2 so, from normal differentiation, we get dy/du = 2u. We also then have u = 4-x. So, again from normal differentiation techniques, we have du/dx=-1.

Using the chain rule gives dy/dx = (2u)*(-1) and if we substitute u=4-x we get

dy/dx = -2(4-x)  = 2x-8    which is the final answer.