Compound Growth Rates

Calculus 1 · Axiom Academy

One relationship — the rate of growth equals the rate times the balance — turns a savings account into the most important curve in calculus. You put 1,000 in an account at 10% a year . The bank can add that interest once a year, or monthly, or every instant — and the more often it compounds, the faster the money grows itself. Pick how often the interest gets added, hit Go, and watch the balance climb deposit by deposit. Compound more often and you finish higher — but never above one ceiling. The bigger the balance, the faster it grows Drag the balance and read off how fast it is earning at that instant. Same 10% rate, but the dollars-per-year keeps climbing — the growth rate is just 10% of whatever you hold right now: rate = r × balance . Why this curve is its own derivative Slide along the continuous-growth curve A = Pe rt . At every point the steepness of the tangent equals r × the height — the curve's slope copies its own value. That self-matching is exactly why e x is the one function that is its own derivative. One rule ran all three: the rate of growth equals r times the amount . Compress the compounding and you reach A = Pe rt ; its slope copies its height, so e x is its own derivative . The very same dA/dt = rA describes populations , radioactive decay (negative r), and a cooling cup of coffee — which is why this one curve shows up everywhere in calculus and science.

This is the written version of the interactive lesson above. See the full Calculus 1 course.