Infinitesimal Change
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- maths partner
We are familiar with finding the gradient of a straight line - we use two points that lie on the line and divide their difference in y by their difference in x:
What about finding the gradient of a curve then? Well, this is a little more tricky.
Firstly we need to be familiar with the notion of a tangent - it's a line which just touchesthe curve at one point.
As you can see in the activity below, as we zoom in more and more, the curve approaches a straight line - in fact it approaches the gradient of the tangent. If we were to keep zooming in an infinite amount of times, eventually the curve would become exactly a straight line. When will this happen? Well when we are looking at an infinitesimal change - that is a change so small that if it were any smaller it would be zero (think of it as the opposite of infinity).
The idea of an infinitesimal lies at the heart of calculus and is what makes it work
However finding the gradient of this tangent is a problem for two reasons:
- The gradient will be different at different points (it's a curve!)
- In order to compute gradient, we need two points - a tangent has just one (by definition - it just touches the curve)
First Principles
Below is a derivation of the formula for Differentiation from First Principles. It's really just using our formula for gradient: