Derivative of a·e^(bx) at a Point

The exponential function e^x is its own derivative: at every point its slope equals its height. That is what makes e the natural base, since it is the one number for which the slope of the curve at x = 0 is exactly 1. Scale and stretch it to a·e^(bx) and the slope becomes b times the height, ab·e^(bx). Worked example: y = 5e^(0.2x) at x = 10. The exponent is 0.2 × 10 = 2, e² = 7.389, the height is 5 × 7.389 = 36.945, and the slope is 0.2 × 36.945 = 7.389. The higher the curve gets, the faster it climbs, which is the whole character of exponential growth.

The classic mistake is to treat e^x like a power and write x·e^(x−1). The power rule is for x in the base with a fixed exponent; here x is in the exponent, and the rule does not apply. A base other than e is handled by rewriting it: 2^x = e^(x ln 2), so enter b = ln 2 = 0.6931 and the slope of 2^x comes out as 2^x ln 2. A negative b gives decay, with a negative slope that shrinks as the quantity does. Any quantity whose rate of change is proportional to the amount present follows this curve: money compounding continuously, a bacterial culture with room to grow, a radioactive sample, the gap between a cup of coffee and the room it is cooling in.