Show, by first principles, that the differential of x^2 is 2x.

First I would draw a Diagram of y=x² on the x and y plane, for x>0. Then label a generic point as (x,y), then noting that as y=x² we can then equate the sane point to (x, x²). Then I would note that another point, further away from the initial point can be labelled as (x+h, (x+h)² ).
I would remind the student that the process of finding a derivative is to find a function for the gradient tangent at any point. I would show the student that using the generic equation for the gradient, we know as y₂-y₁/x₂-x₁. We know two points on this curve, so would tell them to plug these into the equation (as this will give us an estimate for the gradient). Then, I would show graphically that as h gets smaller and smaller, the line gets closer and closer to being a tangent at this point. This introduces the concept of the limit as h approaches zero. I would show them how to plug into the equation and show what happens as h approaches zero for the function, showing that it eventually gives us dy/dx=2x (showing the notation of the different kinds of changes). Then I would ask if there are any questions and then give them a slightly different example.