Defining the Derivative and Using Derivative Notation

Start with slope. The slope between two points on a curve, f(x) and f(x+h), is [f(x+h) - f(x)]/h, the rise over run of a secant line. That fraction is honest but it describes an average rate over a gap of width h. To get the slope at a single point, the instantaneous rate, you shrink h toward zero. You can't just set h = 0, since that gives 0/0, so you take the limit instead. That limit, when it exists, is the derivative: f'(x) = lim(h→0) [f(x+h) - f(x)]/h. The limit is what turns a fraction about two separated points into a statement about one point.

This is why dy/dx is not an ordinary fraction. It's shorthand for a completed limiting process, one single number (or one function of x) that happens to carry fraction-like notation because of where it came from. You can't cross-multiply dy/dx = 3 into dy = 3 dx and treat dy and dx as independent quantities, because there's no dy or dx alive on their own, only the ratio's limit. The same idea can be written with x approaching a directly: f'(a) = lim(x→a) [f(x) - f(a)]/(x - a), where h = x - a has simply been renamed. Both versions describe the same secant-to-tangent squeeze.

This limit definition is the seed for almost everything else in the course. Once you trust that derivatives come from this limit, you can prove the power rule, product rule, and chain rule instead of guessing them. Later, the same difference-quotient idea runs in reverse: instead of shrinking an interval to get an instantaneous rate, the Fundamental Theorem of Calculus builds up infinitely many instantaneous contributions to recover total change, linking derivatives and integrals as two sides of the same limiting process.