Since you already know that the gradient of a straight line joining two points is \( \displaystyle \frac{y_2 - y_1}{x_2 - x_1} \), Isaac Newton and later mathematicians such as Leibniz used the same idea to study curves. They calculated the slope of a chord between two nearby points on a curve and then asked: what happens as those two points come infinitely close? In that limit, the chord becomes the tangent, and its slope becomes what we now call the derivative.
Touch and move the chord as you wish — or watch it turn into the tangent as Q approaches P.
The derivative \( \displaystyle \frac{dy}{dx} \) is the gradient of the chord when the two points that form the chord are brought infinitely close together — i.e. the slope of the tangent.
This simple but powerful idea forms the foundation of the First Principle Method in differentiation. Let’s express this concept step-by-step and see how Newton’s geometric vision becomes a formula.
① What is the slope of the chord PQ?
Simplifying, we get —
(This represents the average rate of change between P and Q.)
② Now tell me: As Q approaches P, what happens to this chord?
③ Therefore,
This is the First Principle of Differentiation — Newton’s way of defining the instantaneous rate of change.