Derivatives of cos x, sin x, e^x, and ln x

These four derivatives are worth memorizing cold, because almost every function later gets built out of them. Picture a point moving around the unit circle at steady speed, with height sin(x) and horizontal position cos(x). The vertical velocity of that point - how fast the height is changing - turns out to equal the horizontal position, cos(x). That's not a coincidence of algebra, it's geometry: when the point is at the top of the circle, its height is barely changing (slope near zero) and cos(x) there is near zero too. When the point crosses the x-axis, height is changing fastest, and cos(x) there is at its max. The graph of cos(x) really is the slope graph of sin(x), point for point.

e^x is stranger and, in a way, simpler: it's defined as the one function whose rate of growth at any point equals its own value there. That's why banks and bacteria both grow 'exponentially' - the bigger the quantity, the faster it grows, and e^x is the purest version of that idea, with growth rate exactly 1 times itself. ln(x), as the inverse of e^x, inherits a derivative of 1/x, which you'll later derive formally using implicit differentiation.

These four rules sit next to the power rule and constant multiple rule as the basic vocabulary of differentiation. Once chain rule arrives, you'll stack these with inside functions - e^3x, sin(x²), ln(5x) - so get the plain versions automatic now.