Differentiation At The Basics

I`m a big advocate of looking at this - given time I always try and fit this into my A level maths lessons. The reason being it gives a real sense of why differentiation works.

Unfortunately on here I am limited and can`t include graphs or nicely formatted formulae, but here I think I can just about get away with the typesetting needed.

A quick analysis here is based on the fact that y=(x squared). A big change is worked out between two points. The gradient between x=1 and x=2 is equal to 3. BUT we know the gradient is constantly changing so this is an average over a large change. This is not sufficient to model changes over the whole curve.

What we need to do is approximate in a space where the curve matches our quick "gradient = change in y over change in x" model as closely as possible. This is easy. If we zoom in on a curve enough, it will begin to look like a straight line. Don't believe me? The earth is curved if seen from space but if you zoom closely enough it appears flat.

so instead we`re going to look at the tiniest change in x, from x to (x+h), where h is tiny. Then if: y = x^2, then

gradient = ((x+h)^2-x^2) / (x+h)-x = x^2 +2xh +h^2 -x^2 / h = 2xh+h^2 / h = h(2x+h) / h = (2x+h)

Now all we do is reduce the size of h until it reaches 0. So over no change in x value whatsoever. If y=x^2 then the gradient dy/dx=2x.

This is so crucial as a starting point and as a method it can be used to explore further differentials. I will always use it as a starting point, if I can though.

Thanks for reading!