Calculus Made Easy

I want to post what I consider to be one of the easiest explanations of simple differentiation, that I have found so far.  It's from the book 'Calculus Made Easy' Thompson, Gardner and it is very straight forward but beautiful at the same time. Just to clarify,  I am using ^2 to mean 'squared'

Here goes:

Starting with a very straight forward expression for a curve

y=x^2

We have the simple task of working out the growth  ratio between Y and X squared.  At a glance, I know that as y increases, x will increase, but I need to know by how much for each value of y.

So if the growth of y is represented as dy and the growth of x is dx, the growth ratio is
dy/dx

If x grows by a little bit, it can be expressed as x+dx, which is x plus a little bit of x.

If y grows, it can be shown as y+dy, which is y plus a little bit of y.

With me so far?

Therefore, the ratio of that growth is found by simply substituting these expressions of growth, into y=x^2.

We do this by replacing y by the quantity y +dy and the quantity x,  by x +dx.

As follows...

y=x^2,  now becomes:

y+dy = (x+dx)^2

Doing the squaring to get rid of the brackets, we get:

y+dy = x^2 + 2x(dx) +(dx)^2

Looking at this, because dx was a very, very small bit of x; by definition, dx squared, is now an extremely small bit of x.  It is so small, in fact, that we can throw it away, and it won't affect our equation too much.

So that leaves,

y+dy = x^2 + 2x(dx)

Now, we know y = x^2, so we can subtract this from both sides, to simplify the equation,

dy = 2x(dx)

We want to know the ratio of dy/dx, so we rearrange the above equation, to give us dy/dx:

dy/dx = 2x

And that, is it!