Hello, wonderful mathematics people. This is Anna Cox from Kellogg Community College. The slope of the curve y=f(x) at the point x naught, f(x) naught is the number m equal the limit as h approaches 0 of f(x naught + h) Minus f(x naught), all divided by h, provided that the limit exists. The tangent line to the curve at p is the line through p with this slope Inhale If we were thinking about our slope formula being y two - y one, over x two - x one This formula truly comes from letting h = x two - x one Inhale So then, h + x one would equal x two Inhale So when we have f(x two) - f(x one) Exhale? All over x two - x one, instead of the x two, we would put in that x one plus h In Hale And instead of the x two minus the x one on the bottom, we put in what it equals. Because it's a slope of a curve, we want to figure out what's happening at the specific point Which is why we put the limit as h goes to 0 The derivative of a function f at a point x naught, denoted f prime (x naught) is f prime (x naught) = the limit as h goes to zero of f( x naught + h) - f(x naught) all over h.