Understanding First Principles
Ever wondered how we actually work out the gradient of a curve at any point? First principles gives you the mathematical foundation behind all derivative calculations.
The key idea is brilliant in its simplicity: if you draw a line between two points on a curve and gradually move those points closer together, the line approaches the tangent to the curve. This tangent's gradient is your derivative.
The first principles formula captures this perfectly: f' = lim[h→0] /h. Here, h represents the tiny distance between your two points, and as h approaches zero, you get the exact gradient.
Quick Tip: Think of first principles as finding the "instantaneous rate of change" - like working out your exact speed at one specific moment rather than your average speed over a journey.
